A unified approach to compatibility theorems on invertible interpolated operators
aa r X i v : . [ m a t h . F A ] A ug A unified approach to compatibility theoremson invertible interpolated operators
I. Asekritova, N. Kruglyak and M. Masty lo
Abstract
We prove the stability of isomorphisms between Banach spaces generated by interpola-tion methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as specialcases, the real and complex methods up to equivalence of norms and also the so-called ± or G and G methods defined by Peetre and Gustavsson–Peetre. This result is usedto show the existence of solution of certain operator analytic equation. A by-product ofthese results is a more general variant of the Albrecht–M¨uller result which states that theinterpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. Weshow applications for positive operators between Calder´on function lattices. We also deriveconnections between the spectrum of interpolated operators. In Banach space theory, operator theory plays a fundamental role. An important part of thistheory is the spectral theory which has applications in many areas of modern analysis andphysics. The study of stability properties of interpolated operators is a central task in abstractinterpolation theory. Motivated by applications in the mentioned areas of analysis, we studystability and the local uniqueness-of-inverse properties of interpolated isomorphisms betweenBanach spaces generated by some general interpolation methods.As usual for a given Banach space X we denote by L ( X ) the Banach space of all boundedlinear operators on X equipped with the standard norm. For basic notation for interpolationtheory, we refer to [3] and [4]. We shall recall that a mapping F : ~ B → B from the category ~ B of all couples of Banach spaces into the category B of all Banach spaces is said to be an interpolation functor if, for any couple ~X := ( X , X ), the Banach space F ( X , X ) is inter-mediate with respect to ~X (i.e., X ∩ X ⊂ F ( ~X ) ⊂ X + X ), and T : F ( X , X ) → F ( Y , Y )for all T : ( X , X ) → ( Y , Y ); here as usual the notation T : ( X , X ) → ( Y , Y ) meansthat T : X + X → Y + Y is a linear operator such that the restrictions of T to thespace X j is a bounded operator from X j to Y j , for both j = 0 and j = 1. An operator T : ( X , X ) → ( Y , Y ) between Banach couples is said to be invertible whenever the restric-tion T | X j : X j → Y j is invertible (i.e., T is an isomorphism of X j onto Y j ) for each j ∈ { , } . Mathematics Subject Classification : Primary 46B70, Secondary 47A13.
Key words and phrases : Interpolation functor, the complex interpolation method, the real interpolationmethod, uniqueness-of-inverses property, spectrum of interpolated operators.The third author was supported by the National Science Centre, Poland, Grant no. 2015/17/B/ST1/00064. ~X = ( X , X ) be a complex Banach couple and T : ( X , X ) → ( X , X ) be an operator.If 0 ≤ α < β ≤ T α := T | [ ~X ] α and T β := T | [ ~X ] β are invertible, then the inverses T − α and T − β do not coincide on X ∩ X in general. Following Zafran [26], an operator T : ~X → ~X issaid to have the uniqueness-of-resolvent (U.R.) property if the restrictions ( T α − λI ) − | X ∩ X and ( T β − λI ) − | X ∩ X coincide for all α , β ∈ [0 ,
1] and λ / ∈ σ ( T α ) ∪ σ ( T β ).Ransford [24] introduced a weaker property; an operator T : ~X → ~X satisfies the localuniqueness-of-resolvent (local U.R.) condition, if for all α ∈ (0 ,
1) and λ / ∈ σ ( T α ), there existsa neighbourhood U ⊂ (0 ,
1) of α such that ( T θ − λI ) − exists and ( T θ − λI ) − | X ∩ X agreeswith ( T α − λI ) − | X ∩ X for all θ ∈ U . Albrecht and M¨uller proved in [1] that this condition isalways fulfilled. This follows immediately from the following result (see [1, Theorem 4] whichstates: If ( X , X ) is a complex Banach couple, T : ( X , X ) → ( X , X ) and T α : [ X , X ] α → [ X , X ] α is invertible for some α ∈ (0 , , then there exists a neighbourhood U ⊂ (0 , of α such that T θ is invertible and T − θ agrees with T − α on X ∩ X for any θ ∈ U . Our aim is to provide a unified general approach to abstract compatibility theorems ofstronger type than Albrecht–M¨uller result for operators between Banach spaces generated byabstract interpolation methods. To do this we introduce a new key notion of a stable family { F θ } θ ∈ (0 , of interpolation functors (for an exact definition we refer to Section 4) and provethat certain class of interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg in[8] are stable. In particular, we prove that the Calder´on complex family { [ · ] θ } θ ∈ (0 , as well asthe Lions–Peetre real family { ( · ) θ,q } θ,q for any 1 ≤ q ≤ ∞ of interpolation functors are stablefamilies.The fundamental theorems on a stable family { F θ } θ ∈ (0 , of interpolation functors arethe following main results of this paper true for the restrictions T θ := T | F θ ( X ,X ) from F θ ( X , X ) to F θ ( Y , Y ) of any linear bounded operator T : ( X , X ) → ( Y , Y ) betweenBanach couples: If T θ ∗ : F θ ∗ ( X , X ) → F θ ∗ ( Y , Y ) is invertible for some θ ∗ ∈ (0 , , then there exists a neigh-bourhood U ⊂ (0 , of θ ∗ such that T θ is invertible, the inverse T − θ agrees with T − θ ∗ on Y ∩ Y ( i.e., T − θ ( y ) = T − θ ∗ ( y ) for all y ∈ Y ∩ Y ) , and the following estimate holds : k T − θ k F θ ( Y ,Y ) → F θ ( X ,X ) ≤ k T − θ ∗ k F θ ∗ ( Y ,Y ) → F θ ∗ ( X ,X ) for all θ ∈ U .
As a consequence, the set of all θ ∈ (0 , for which T θ is invertible is an open subset of (0 , . We note that in addition we describe more precisely the mentioned above neighbourhood U ⊂ (0 , { F θ } , which satisfies thereiteration condition, we prove a subtle compatibility result which states: If I ⊂ (0 , is an open interval of invertibility of T ( i.e., such that T θ is invertible for all θ ∈ I ) ,then for any θ , θ ′ ∈ I the inverse operators T − θ and T − θ ′ agree on F θ ( Y , Y ) ∩ F θ ′ ( Y , Y ) . Among several motivations for studying compatibility problems are important applicationsto PDE’s. It seems the roots for these problems are in Calder´on paper [6] in which it is provedthat if (Ω , Σ , µ ) is a measure space and T : L p ( µ ) → L p ( µ ) is a bounded operator for 1 < p < ∞ ,2hich is invertible for p = 2, then T is also invertible when 2 − ε < p < ε , for some small ε >
0. In fact careful analysis of Calder´on’s proofs gives the compatibility of inverses, i.e., thereexists some small ε > p, q ∈ (2 − ε, ε ), the inverse T − considered on thespace L p ( µ ) is compatible with T − considered on L q ( µ ) when both operators are restrictedto L p ( µ ) ∩ L q ( µ ). It was shown in [23] very useful application for solvability of the Dirichletproblem with data in L p ( ∂ Ω) for the biharmonic equation ∆ u = 0 in Ω, u = f and ∂u/∂n = g on ∂ Ω, in a bounded Lipschitz domain Ω ⊂ R n .It is worth pointing out that in the remarkable paper [13] by Kalton–Mayaboroda–Mitreathere are shown applications of compatibility results for the variants of the Dirichlet problemas well as the Neumann problem for the Laplacian in L p ( ∂ Ω)-spaces in the case of unboundeddomain Ω above the graph of a real-valued Lipschitz function defined in R n − .We conclude by noting that using the well known technics to the mentioned above typePDE’s, our compatibility results can be applied to other methods than the complex. In partic-ular, applying the real method, we would get variants of the Dirichlet problem as well as theNeumann problem for the Laplacian in Lorentz L p,q ( ∂ Ω)-spaces.Throughout the paper we shall require considerable notation. If X and Y are Banach spacessuch that X ⊂ Y and the inclusion map id : X → Y is bounded, then we write X ֒ → Y . Forsimplicity of notation, we write X ∼ = Y whenever X = Y , with equality of norms. We introduce the basic notations and definitions to be used throughout this work. We will usecomplex methods of interpolation introduced by Calder´on in his fundamental paper [5].Let S := { z ∈ C ; 0 < Re z < } be an open strip on the plane. For a given θ ∈ (0 , ~X = ( X , X ) we denote by F ( ~X ) the Banach space of all bounded continuousfunctions f : ¯ S → X + X on the closure ¯ S that are analytic on S , and R ∋ t f ( j + it ) ∈ X j is a bounded continuous function, for each j ∈ { , } , and endowed with the norm k f k F ( ~X ) = max j =0 , sup t ∈ R k f ( j + it ) k X j . The lower complex interpolation space is defined by [ ~X ] θ := { f ( θ ); f ∈ F ( ~X ) } and is endowedwith the quotient norm. This definition is slightly different from those in [3, 5], however itgives the same interpolation spaces (see, e.g., [5]). We recall that in the original definition it isrequired in addition that f ∈ F ( ~X ) satisfieslim | t |→∞ k f ( j + it ) k X j = 0 , j ∈ { , } . We also recall the basic constructions and results of [8] which we will use here, and we refer tothis paper for more details. Let
Ban be the class of all Banach spaces over the complex field.A mapping X : Ban → Ban is called a pseudolattice, or a pseudo - Z - lattice , if(i) for every B ∈ Ban the space X ( B ) consists of B valued sequences { b n } = { b n } n ∈ Z modelled on Z ;(ii) whenever A is a closed subspace of B it follows that X ( A ) is a closed subspace of X ( B );(iii) there exists a positive constant C = C ( X ) such that, for all A , B ∈ Ban and all boundedlinear operators T : A → B and every sequence { a n } ∈ X ( A ), the sequence { T a n } ∈ X ( B ) and3atisfies the estimate k{ T a n }k X ( B ) ≤ C k T k A → B k{ a n }k X ( A ) ;(iv) k b m k B ≤ k{ b n }k X ( B ) for each m ∈ Z , all { b n } ∈ X ( B ) and all Banach spaces B .For every Banach couple ~B = ( B , B ) and every Banach couple of pseudolattices ~ X =( X , X ), let J ( ~ X , ~B ) be the Banach space of all B ∩ B valued sequences { b n } such that { e jn b n } ∈ X ( B j ) ( j = 0 , k{ b n }k J ( ~ X , ~B ) = max (cid:8) k{ b n }k X ( B ) , k{ e n b n }k X ( B ) (cid:9) . Following [8], for every s in the annulus A := { z ∈ C ; 1 < | z | < e } , we define the Banachspace ~B ~X,s to consist of all elements of the form b = P n ∈ Z s n b n (convergence in B + B with { b n } ∈ J ( ~ X , ~B ), equipped with the norm k b k ~B ~ X ,s = inf (cid:26) k{ b n }k J ( ~ X , ~B ) ; b = X n ∈ Z s n b n (cid:27) . It is easy to check that the map ~B ~B ~ X ,s is an interpolation functor.We will consider mainly couples ~ X = ( X , X ) of Banach pseudolattices, which are transla-tion invariant, i.e., such that any Banach space B we have (cid:13)(cid:13) { S k ( { b n } n ∈ Z (cid:9)(cid:13)(cid:13) X j ( B ) = (cid:13)(cid:13) { b n } n ∈ Z (cid:13)(cid:13) X j ( B ) for all { b n } n ∈ Z ∈ X j ( B ), each k ∈ Z and j ∈ { , } . Here and in what follows S denote theleft-shift operator on two-sided (vector valued) sequences defined by S { b n } = { b n +1 } .Following [8] ~ X = ( X , X ) is said to be a rotation invariant Banach couple of pseudolatticeswhenever the rotation map { b n } n ∈ Z
7→ { e inτ b n } n ∈ Z is an isometry of X j ( B ) onto itself for everyreal τ and every Banach space B .The following useful lemma is obvious, but we include a proof. Lemma 2.1.
Let ~ X = ( X , X ) be a Banach couple of rotation invariant pseudolattices. Then,for every Banach couple ~B = ( B , B ) and all s ∈ A , we have (i) If f ∈ F ~ X ( ~B ) , then f ( s ) ∈ ~B ~ X , | s | ;(ii) If x ∈ ~B ~ X , | s | , then there exists f ∈ F ~ X ( ~B ) such that f ( s ) = x ;(iii) ~B ~ X ,s ∼ = ~B ~ X , | s | .Proof. (i). Let f ∈ F ~ X ( ~B ). Then there exists { b n } n ∈ Z ∈ J ( ~ X , ~B ) such that f ( z ) = P n ∈ Z z n b n for all z ∈ A (convergence in B + B ). Define e f by e f ( z ) = f ( ze iϕ ) for all z ∈ A , where ϕ := Arg s . Then e f ( z ) = X n ∈ Z z n e inϕ b n , z ∈ A . k{ e − inϕ b n }k J ( ~ X , ~B ) = k{ b n }k J ( ~ X , ~B ) and so e f ∈ F ~ X ( ~B ). Since f ( s ) = e f ( | s | ) ∈ ~B ~ X , | s | , f ( s ) ∈ ~B ~ X , | s | and this proves (i).(ii). Let x ∈ ~B ~ X ,s . Then there exists e f ∈ F ~ X ,s ( ~B ) with b = { b n } ∈ J ( ~ X , ~B ) such that e f ∈ F ~ X ( ~B ) and e f ( | s | ) = x , where e f ( z ) = X n ∈ Z z n b n , z ∈ A . Define f by f ( z ) = e f ( ze − iϕ ) for all z ∈ A . Our hypothesis gives that f ∈ F ~ X ( ~B ). Combiningthe above facts yields f ( s ) = e f ( | s | ) = x and this proves (ii).(iii). It is enough to observe that the proofs of (i) and (ii) yields k f k F ~ X ( ~B ) = k e f k F ~ X ( ~B ) , f ∈ F ~ X ( ~B ) . We note that the above lemma shows if ~ X = ( X , X ) is a Banach couple of rotationinvariant pseudolattices, then for any s = e θ + iϕ with θ ∈ (0 ,
1) and ϕ ∈ [0 , π ), we have that ~B ~ X ,s ∼ = ~B ~ X ,e θ for any Banach couple ~B .We point out that concerning interpolation methods the idea of [8] was to show that a largefamily of interpolation methods have a suitable complex analytic structure that could be usedfor methods that apriori do not seem to have one. This essential fact is deeply used in ourpaper. Note that with the right choices of pseudolattice couples ( X , X ), we recover the classicalmethods of interpolation (see [8] for more details). In particular let s = e θ with 0 < θ <
1. If X = X = ℓ p with 1 ≤ p ≤ ∞ , the space ~B ~ X ,s coincides with the Lions–Peetre real J -methodspace ~B θ,p ; J (see, e.g., [18, p. 41] where this space is denoted by s ( p, θ, B ; p, θ − , B ).It is well known that ( B , B ) θ,p ; J = ( B , B ) θ,p up to equivalence of norms (see [3, Chap. 3]),where ( B , B ) θ,p is the K -method space endowed with the norm k b k θ,p := (cid:18) ˆ ∞ (cid:0) t − θ K ( t, b ; ~B ) (cid:1) p dtt (cid:19) /p , ≤ p < ∞ . For θ ∈ [0 ,
1] and p = ∞ the real interpolation space ~B θ, ∞ is defined to be a space of all b ∈ B + B endowed with the norm k b k θ, ∞ := sup t> t − θ K ( t, b ; ~B ) . Here as usual for any Banach couple ~X = ( X , X ) the Peetre K -functional is defined by K ( t, x ; ~X ) = K ( t, x ; X , X ) := inf {k x k X + k x k X ; x + x = x } , t > . Let X be a Banach space intermediate with respect to a Banach couple ~X = ( X , X ). The Gagliardo completion or relative completion of X with respect to ~X is the Banach space X c of all limits in X + X of sequences that are bounded in X and endowed with the norm k x k X c = inf { sup k ≥ k x k k X } , where the infimum is taken over all bounded sequences { x k } in X whose limit in X + X equals x . 5e will use without any references the well-known fact (see [4, Lemma 2.2.30]) that for anyBanach couple ( X , X ) we have( X ) c ∼ = ( X , X ) , ∞ , ( X ) c ∼ = ( X , X ) , ∞ . If ~ X = ( F C, F C ), then ~B ~ X ,s coincides, to within equivalence of norms, with the Cader´oncomplex method space [ ~B ] θ = [ B , B ] θ (see [7]). If ~ X = ( U C, U C ), then ~B ~ X ,s is the ± method space h ~B i θ ∼ = h B , B i θ (see [22, p. 176]). If we replace U C by W U C , we obtain theGustavsson–Peetre variant of h B , B i θ which is denoted by h ~B ; θ i (see [10, p. 45], [12]). Throughout the paper, for an operator T : ~X → ~Y between Banach couples and every ω ∈ A ,we often denote by T ω the restriction T | ~X ~ X ,ω : ~X ~ X ,ω → ~Y ~ X ,ω . For simplicity of notation, wewrite T θ instead of T e θ for any θ ∈ (0 , Theorem 3.1.
Let ~ X = ( X , X ) be a Banach couple of translation invariant pseudolattices andlet T : ~X → ~Y be an operator between complex Banach couples. Assume that T : ~X ~ X ,s → ~Y ~ X ,s is invertible for some s ∈ A . Then T ω : ~X ~ X ,ω → ~Y ~ X ,ω is invertible for all ω in an openneighbourhood W = { ω ∈ A ; | ω − s | < r } of s in A with r = (cid:2) δ ( s ) (cid:0) k T k ~X → ~Y k T − k ~Y ~ X ,s → ~X ~ X ,s (cid:1)(cid:3) − , where δ ( s ) = max (cid:8) ( | s |− − , ( e − | s | ) − (cid:9) . Moreover the following upper estimate for the normof T ω holds, (cid:13)(cid:13) T − ω (cid:13)(cid:13) ~Y ~ X ,ω → ~X ~ X ,ω ≤ (cid:13)(cid:13) T − s (cid:13)(cid:13) ~Y ~ X ,s → ~X ~ X ,s , ω ∈ W. In the case when ~ X = ( X , X ) is a couple of translation and rotation invariant pseudolatticeswe obtain the following variant of Albrechr–Miller result. Theorem 3.2.
Let ~ X = ( X , X ) be a couple of translation and rotation invariant pseudolatticesand let T : ~X → ~Y be an operator between complex Banach couples. Assume that T θ ∗ : ~X ~ X ,e θ ∗ → ~Y ~ X ,e θ ∗ is invertible for some θ ∗ ∈ (0 , . Then T θ : ~X ~ X ,e θ → ~Y ~ X ,e θ is invertible for all θ in anopen neighbourhood I = { θ ∈ (0 , | θ − θ ∗ | < ε } of θ ∗ with ε = (cid:2) eη ( θ ∗ ) (cid:0) k T k ~X → ~Y k T − k ~Y ~ X ,eθ ∗ → ~X ~ X ,e θ ∗ (cid:1)(cid:3) − , where η ( θ ∗ ) = max (cid:8) ( e θ ∗ − − , ( e − e θ ∗ ) − (cid:9) . Moreover, T − θ agrees with T θ ∗ on Y ∩ Y and (cid:13)(cid:13) T − θ (cid:13)(cid:13) ~Y ~ X ,eθ → ~X ~ X ,eθ ≤ (cid:13)(cid:13) T − θ ∗ (cid:13)(cid:13) ~Y ~ X ,eθ ∗ → ~X ~ X ,eθ ∗ for any θ ∈ I. To prove this theorem we will need some preliminary results. We start our investigationwith the following a more precise cancellation principle (cf. [8]. We note that careful analysisof the proof Lemma 3.1 in [8] gives a key Lipschitz estimate with a constant depending onparameter s ∈ A , but not on the couple of translation invariant pseudolattices. Since thisestimate is essential in our study, we include a proof for readers’ convenience.6 emma 3.3. Let ~ X be a couple of translation invariant pseudolattices and let ~B = ( B , B ) bea Banach couple. Let the sequence { f n } n ∈ Z be an element of J ( ~ X , ~B ) and let f : A → B + B be the analytic function defined by f ( z ) = P n ∈ Z z n f n . Suppose that f ( s ) = 0 for some s ∈ A and let g : A → B + B be given by g ( s ) = f ′ ( s ) and g ( z ) = f ( z ) / ( z − s ) for all z ∈ A \ { s } .Then, g ∈ F ~ X ( ~B ) and the Laurent expansion of g in A , g ( z ) = P n ∈ Z z n g n for all A , satisfies { g n } n ∈ Z ∈ J ( ~ X , ~B ) and k{ g n }k J ( ~ X , ~B ) ≤ δ ( s ) k{ f n }k J ( ~ X , ~B ) , where δ ( s ) = max (cid:8) ( | s | − − , ( e − | s | ) − (cid:9) .Proof. Let f ∈ F ~ X ( ~B ) be such that f ( s ) = 0. We define g ( z ) = ( f ( z ) z − s , if z = sf ′ ( z ) , if z = s . Clearly g : A → B + B is analytic. Let f ( z ) = P n ∈ Z f n z n for all z ∈ A , where { f n } ∈ J ( ~ X , ~B ).We claim that the Laurent expansion of g in A satisfy the required properties. Because of theuniqueness of the Laurent expansion, it is enough to show that g ( z ) = P n ∈ Z z n g n for all | s | < | z | < e , and moreover that { g n } ∈ J ( ~ X , ~B ) satisfies the desired estimate.Fix z ∈ A such that | z | > | s | . Combining the absolute convergence of series, we have g ( z ) = 1 z (cid:0) − sz (cid:1) X n ∈ Z f n z n = X k ≥ z (cid:16) sz (cid:17) k X n ∈ Z f n z n = X n ∈ Z X k ≥ z n − k − s k f n = X m ∈ Z (cid:16) X k ≥ s k f m + k +1 (cid:17) z m . Since f ( s ) = 0, we get that P n ≥ k s n f n = − P n Theorem 3.4. Suppose that H : U → V maps U j to V j for each j ∈ { , } , and the quotientoperator H : U/U → V /V is invertible. If max { dist( U , U ) , dist( V , V ) } < (cid:0) k H k U → V k H − k V/V → U/U (cid:1) , hen the quotient operator H : U/U → V /V is invertible. Moreover the upper estimate forthe norm of H is given by k H − k V/V → U/U ≤ k H − k V/V → U/U . Let ~ X be a Banach couple of pseudolattices, ~B a Banach couple, and let “dist” be a distancedefined on closed subspaces of the space F ~ X ( ~B ), and let s , ω ∈ A . Then we define ρ (cid:0) ~B ~ X ,s , ~B ~ X ,ω (cid:1) := dist (cid:0) N s ( ~B ) , N ω ( ~B ) (cid:1) . The following variant of a result from [15] is relevant to our purposes. Theorem 3.5. Let ~B be a complex Banach couple. Then, for all s ∈ A , dist (cid:0) ( · ) ~ X ,s , ( · ) ~ X ,ω (cid:1) := sup ~B ~ B ρ (cid:0) ~B ~ X ,s , ~B ~ X ,ω (cid:1) ≤ δ ( s ) | ω − s | , ω ∈ A , where δ ( s ) = max (cid:8) ( | s | − − , ( e − | s | ) − (cid:9) .Proof. We have ρ (cid:0) ~B ~ X ,s , ~B ~ X ,ω (cid:1) = sup k f k F ~ X ( ~B ) ≤ (cid:12)(cid:12) ρ (cid:0) f, N s ( ~B ) (cid:1) − ρ (cid:0) f, N ω ( ~B ) (cid:1)(cid:12)(cid:12) = sup k f k F ~ X ( ~B ) ≤ (cid:12)(cid:12)(cid:12)(cid:13)(cid:13) f + N s ( ~B ) (cid:13)(cid:13) F ~ X ( ~B ) /N s ( ~B ) − (cid:13)(cid:13) f + N ω ( ~B ) (cid:13)(cid:13) F ~ X ( ~B ) /N ω ( ~B ) (cid:12)(cid:12)(cid:12) = sup k f k F ~ X ( ~B ) ≤ (cid:12)(cid:12)(cid:12) k f ( s ) k ~B ~ X ,s − k f ( ω ) k ~B ~ X ,ω (cid:12)(cid:12)(cid:12) . Let f ∈ F ~ X ( ~B ) be such that k f k F ~ X ( ~B ) ≤ 1, and let x = f ( s ). Given ε > f x ∈ F ~ X ( ~B )such that f x ( s ) = x, k f x k F ~ X ( ~B ) ≤ k x k ~B ~ X ,s + ε. In particular we have k f x k F ~ X ( ~B ) ≤ ε . Since f ( s ) − f x ( s ) = 0, it follows from Lemma 3.3that the function h defined by h ( s ) = f ′ ( s ) − f ′ x ( s ) and h ( z ) = f ( z ) − f x ( z ) z − s , z ∈ A \ { s } is in F ~ X ( ~B ), and k h k F ~ X ( ~B ) ≤ δ ( s )for some positive constant δ ( s ) ≤ max (cid:8) ( | s | − − , ( e − | s | ) − (cid:9) .Now observe that f ( ω ) − f x ( ω ) = ( ω − s ) h ( ω ) , ω ∈ A and so k f ( ω ) − f x ( ω ) k ~B ~ X ,ω ≤ | ω − s |k h ( ω ) k ~B ~ X ,ω ≤ δ ( s ) | ω − s | . ω ∈ A , k f ( ω ) k ~B ~ X ,ω ≤ k f x ( ω ) k ~B ~ X ,ω + δ ( s ) | ω − s |≤ k f x k F ~ X ( ~B ) + δ ( s ) | ω − s |≤ k x k ~B ~ X ,s + δ ( s ) | ω − s | + ε ≤ k f ( s ) k ~B ~ X ,s + δ ( s ) | ω − s | + ε. Since ε is arbitrary, we get (cid:12)(cid:12)(cid:12) k f ( ω ) k ~B ~ X ,ω − k f ( s ) k ~B ~ X ,s (cid:12)(cid:12)(cid:12) ≤ δ ( s ) | ω − s | , ω ∈ A , and this completes the proof.We are ready for the proof Theorem 3.1. Proof of Theorem . For ω ∈ A define the operator e T ω : F ~ X ( ~X ) /N ω ( ~X ) → F ~ X ( ~Y ) /N ω ( ~Y ) by e T w ( f + N ω ( ~X )) = e T f + N ω ( ~Y ) , f + N ω ( ~X ) ∈ F ~ X ( ~X ) /N ω ( ~X ) , where e T : F ~ X ( ~X ) → F ~ X ( ~Y ) is the operator given by e T f ( z ) = T ( f ( z )) , f ∈ F ~ X ( ~X ) , z ∈ A . We note that k e T k F ~ X ( ~X ) →F ~ X ( ~Y ) ≤ k T k ~X → ~Y = max j =0 , k T k X j → Y j and k e T ω k F ~ X ( ~X ) /N ω ( ~X ) →F ~ X ( ~Y ) /N ω ( ~Y ) = k T k ~X ~ X ,ω → ~Y ~ X ,ω . ( ∗ )Now we fix s ∈ A . Then, from Theorem 3.5, we conclude that for δ ( s ) = max (cid:8) ( | s | − − , ( e − | s | ) − (cid:9) we havedist (cid:0) ( · ) ~ X ,s , ( · ) ~ X ,ω (cid:1) ≤ δ ( s ) | ω − s | , ω ∈ A . Let W := { ω ∈ A ; | ω − s | < r } be an open neighbourhood of s in A with r = (cid:0) δ ( s ) + 2 δ ( s ) k T − k ~Y ~ X ,s → ~X ~ X ,s k T k ~X → ~Y (cid:1) − . Then, we havedist (cid:0) ( · ) ~ X ,s , ( · ) ~ X ,ω (cid:1) < (cid:0) k T k ~X → ~Y k T − k ~Y ~ X ,s → ~X ~ X ,s (cid:1) , ω ∈ W. Combining the above with Theorem 3.4 applied to the Banach spaces U = F ~ X ( ~X ), V = F ~ X ( ~Y ),closed subspaces U = N s ( ~X ), U = N ω ( ~X ) ⊂ U and V = N s ( ~Y ), V = N ω ( ~Y ) ⊂ V with ω ∈ W , and operators H = e T , H = e T s , H = e T ω , we obtain the desired statement for shownabove open neighborhood W ⊂ A of s .To get the required estimate for the norm of T − ω for all ω ∈ W , we first observe thatfollowing the above notation, it follows from equality ( ∗ ) that k H k U → V = k T k ~X → ~Y and k H − k V/V → U/U = k T − ω k ~Y ~ X ,ω → ~X ~ X ,ω , k H − k V/V → U/U = k T − ω k ~Y ~ X ,s → ~X ~ X ,s . 10y Theorem 3.5, for all ω ∈ W , we havemax { dist( U , U ) , dist( V , V ) } ≤ dist (cid:0) ( · ) ~ X ,s , ( · ) ~ X ,ω (cid:1) < k T k ~X → ~Y k T − k ~Y ~ X ,s → ~X ~ X ,s ) . To finish, we apply Theorem 3.4 to get the required estimate.We isolate the following simple proposition for further reference. Proposition 3.6. Let ~ X = ( X , X ) be a couple of pseudolattices and let ~Y be a Banach couple.Then, for every ω ∈ A , the operator V ω : F ~ X ( ~Y ) → F ~ X ( ~Y ) defined by ( V ω f )( z ) = ( ω − z ) f ( z ) , f ∈ F ~ X ( ~Y ) , z ∈ A is injective and it has closed range with R ( V ω ) = N ω ( ~Y ) .Proof. We first remark that our hypothesis on ~ X yields that a function A ∋ z zf ( z ) ∈ F ~ X ( ~Y )for any f ∈ F ~ X ( ~Y ). Thus the domain D ( V ω ) = F ~ X ( ~Y ). Clearly, V ω f = 0 for f ∈ F ~ X ( ~Y ) impliesthat f ( z ) = 0 for all z ∈ A \ { ω } and whence f = 0 by continuity of f .It is obvious that the range satisfies R ( V ω ) ⊂ { g ∈ F ~ X ( ~Y ); g ( ω ) = 0 } = N ω ( ~Y ) . To show the reverse inclusion let g ∈ F ~ X ( ~Y ) with g ( ω ) = 0. It follows from Lemma 3.3 thatthere exists a function f ∈ F ~ X ( ~Y ) such that g ( z ) = ( ω − z ) f ( z ) , z ∈ A . Thus, we get that g = V ω f and so the desired equality R ( V ω ) = N ω ( ~Y ) holds. Since N ω ( ~Y ) isa closed subspace in F ~ X ( ~Y ), the proof is complete.We prove a lemma which will play a key role in the proof of the main result, Theorem 3.2.In the proof we will use some methods from [1, Theorem 4]. We recall that if S : X → Y is a bounded linear operator between Banach spaces, then, the so called lower bound of S isdefined by γ ( S ) = inf {k Sx k Y ; x ∈ X, k x k X = 1 } . It is obvious that γ ( S ) > S is injective and the range R ( S ) of S is a closedsubspace in Y . Lemma 3.7. Let ~ X be a couple of pseudolattices and let ~X = ( X , X ) , ~Y = ( Y , Y ) becomplex Banach couples, T : ~X → ~Y and s ∈ A . Assume that T : ~X ~ X ,s → ~Y ~ X ,s is invertible.Then, there exists an open neighborhood U ⊂ A of s such that, for all k ∈ F ~ X ( ~Y ) , there existanalytic functions g : U → F ~ X ( ~X ) and h : U → F ~ X ( ~Y ) such that, for all ω ∈ U , T ( g ( ω )( z )) + ( ω − z ) h ( ω )( z ) = k ( z ) , z ∈ A . Proof. From Proposition 3.6, it follows that the injective operator V ω : F ~ X ( ~Y ) → F ~ X ( ~Y ) givenfor every ω ∈ A by V ω f ( z ) = ( ω − z ) f ( z ) , f ∈ F ~ X ( ~Y ) , z ∈ A . R ( V ω ) = N ω ( ~Y ). Thus, the lower bound γ ( V ω ) > ω ∈ A . Since V ω = ( ω − s ) id F ~ X ( ~Y ) + V s , ω ∈ A , A ∋ ω V ω ∈ L (cid:0) F ~ X ( ~Y ) , F ~ X ( ~Y ) (cid:1) is an analytic function.We shall adopt notations from Theorem 3.1. Thus we will consider operators e T : F ~ X ( ~X ) →F ~ X ( ~Y ) and e T ω : F ~ X ( ~X ) /N ω ( ~X ) → F ~ X ( ~Y ) /N ω ( ~Y ), where ω ∈ A .We note that k e T k F ~ X ( ~X ) →F ~ X ( ~Y ) ≤ max j =0 , k T k X j → Y j and k e T ω k F ~ X ( ~X ) /N ω ( ~X ) →F ~ X ( ~Y ) /N ω ( ~X ) = k T k ~X ~ X ,ω → ~Y ~ X ,ω . Let c and c be positive constants such that c > k T − s k and c > (cid:0) c k e T k (cid:1) γ ( V s ) − , where, for simplicity of notation, we let k T − s k = k T − k ~Y ~ X ,s → ~X ~ X ,s , k e T k = k e T k F ~ X ( ~X ) →F ~ X ( ~Y ) .It follows from Theorem 3.1 that there exists an open neighbourhood W ⊂ A of s such that T ω : ~X ~ X ,ω → ~X ~ X ,ω is invertible for all ω ∈ W .We claim that an open neighborhood U ⊂ A of s given by U := (cid:8) ω ∈ A ; | ω − s | < c − (cid:9) ∩ W satisfies the required statements, i.e., there exist analytic function g : U → F ~ X ( ~X ) and h : U →F ~ X ( ~Y ) such that e T g ( ω ) + V ω h ( ω ) = k, ω ∈ U. To see this fix k ∈ F ~ X ( ~Y ) and observe that, if g ( ω ) = P ∞ n =0 g n ( ω − s ) n and h ( ω ) = P ∞ n =0 h n ( ω − s ) n are the Taylor expansions of g and h about s , then solution of the required equation e T g ( ω ) + V ω h ( ω ) = k, ω ∈ U with g and h in the form given above reduces to solution of the following recurrence equationsgenerated by the sequences { g n } ⊂ F ~ X ( ~X ) and { h n } ⊂ F ~ X ( ~Y ) of Taylor’s coefficients of g and h , respectively e T g + V s h = k, e T g n + V s h n = − h n − , n ∈ N such that the both series P ∞ n =0 g n ( ω − s ) n and h ( ω ) = P ∞ n =0 h n ( ω − s ) n converge in U .Our hypothesis on invertibility of T s : ~X ~ X ,s → ~Y ~ X ,s implies that e T s : F ~ X ( ~X ) /N s ( ~X ) → F ~ X ( ~Y ) /N s ( ~Y )is also invertible. Thus, there exists f ∈ F ~ X ( ~X ) such that e T s (cid:0) f + N s ( ~X ) (cid:1) = k + N s ( ~Y ) . Combining with k e T − ω k = k T − k ~Y ~ X ,ω → ~X ~ X ,ω for all ω ∈ A , we conclude that (cid:13)(cid:13) f + N s ( ~X ) (cid:13)(cid:13) F ~ X ( ~X ) /N s ( ~X ) = (cid:13)(cid:13)f T s − (cid:0) e T s (cid:0) f + N s ( ~X ) (cid:1)(cid:1)(cid:13)(cid:13) F ~ X ( ~X ) /N s ( ~X ) ≤ k T − s k (cid:13)(cid:13) k + N s ( ~Y ) (cid:13)(cid:13) F ~ X ( ~Y ) /N s ( ~Y ) ≤ k T − s k k k k F ~ X ( ~Y ) . ε = c k T − s k − 1. Then, there exists f ∈ N s ( ~X ) such that k f − f k F ~ X ( ~X ) ≤ (1 + ε ) (cid:13)(cid:13) f + N s ( ~X ) (cid:13)(cid:13) F ~ X ( ~X ) /N s ( ~X ) = c k T − s k (cid:13)(cid:13) f + N s ( ~X ) (cid:13)(cid:13) F ~ X ( ~X ) /N s ( ~X ) ≤ c k k k F ~ X ( ~Y ) . Hence for g := f − f ∈ F ~ X ( ~X ), we have g + N s ( ~X ) = f + N s ( ~X ) and k g k F ~ X ( ~X ) ≤ c k k k F ~ X ( ~Y ) . Clearly this yields (cid:0) by e T f ( s ) = 0 and g + N s ( ~X ) = f + N s ( ~X ) (cid:1) k − e T g ∈ N s ( ~Y ) , and k e T g − k k F ~ X ( ~Y ) ≤ k k k F ~ X ( ~Y ) (cid:0) c k e T k (cid:1) . ( ∗ )We claim that there exists h ∈ F ~ X ( ~Y ) such that V s h = k − e T g , k h k F ~ X ( ~Y ) ≤ c k k k F ~ X ( ~Y ) . To see this observe that, for all h ∈ F ~ X ( ~Y ), we have γ ( V s ) k h k F ~ X ( ~Y ) ≤ k V s h k F ~ X ( ~Y ) . According to Proposition 3.6, we can find (cid:0) by R ( V s ) = N s ( ~Y ) (cid:1) h ∈ F ~ X ( ~Y ) such that V s h = k − e T g . Then by estimate ( ∗ ), one has k h k F ~ X ( ~Y ) ≤ γ ( V s ) k V s h k F ~ X ( ~Y ) ≤ c c k e T k k k − e T g k F ~ X ( ~Y ) ≤ c k k k F ~ X ( ~Y ) . In consequence, we deduce that the claim holds for h .Similarly we find g ∈ F ~ X ( ~X ) and h ∈ F ~ X ( ~Y ) such that e T g + V s h = − h and k g k F ~ X ( ~X ) ≤ c c k k k F ~ X ( ~Y ) , k h k F ~ X ( ~Y ) ≤ c k k k F ~ X ( ~Y ) . Now continuing the process, we construct sequences { g n } ⊂ F ~ X ( ~X ), { h n } ⊂ F ~ X ( ~Y ) such that,for each n ∈ N we have k g n k F ~ X ( ~X ) ≤ c c n − k k k F ~ X ( ~Y ) , k h n k F ~ X ( ~Y ) ≤ c n k k k F ~ X ( ~Y ) . This implies that the functions g : U → F ~ X ( ~X ) and h : U → F ~ X ( ~Y ), given by g ( ω ) = ∞ X n =0 g n ( ω − s ) n , h ( ω ) = ∞ X n =0 h n ( ω − s ) n , ω ∈ U are analytic in U and satisfy the desired statement.13ow we are ready to proof of Theorem 3 . Proof. For a fixed y ∈ Y ∩ Y let k be a constant function given by k ( z ) = y for all z ∈ A .Since k ∈ F ~ X ( ~Y ), it follows from Lemma 3.7 that there exist an open neighborhood U ⊂ A of s and analytic functions g : U → F ~ X ( ~X ), h : U → F ~ X ( ~Y ) such that, for all ω ∈ U and all z ∈ A ,we have T ( g ( ω )( z )) + ( ω − z ) h ( ω )( z ) = y. Define a function e g : U → X + X by e g ( ω ) = g ( ω )( ω ) , ω ∈ U. Then e g is analytic in U and T ( e g ( ω )) = y by the above formula. Further, g ( ω ) ∈ F ~ X ( ~X ) implies e g ( ω ) ∈ ~X ~ X ,ω = ~X ~ X , | ω | (by Lemma 2.1 (iii)). Since T ω : ~X ~ X , | ω | → ~Y ~ X , | ω | is invertible for all ω ∈ U , e g ( ω ) = T − | ω | ( y ) , ω ∈ U. In particular this implies that the analytic function e g is constant on an open arc of the circlewith the center at 0 and radius | s | which is contained in U . Thus e g is constant in U by theuniqueness theorem. Hence T − ω y is independent of ω ∈ U . To finish the proof it is enough tocombine an obvious inequality, | e θ − e θ ∗ | ≤ e | θ − θ ∗ | , θ, θ ∗ ∈ (0 , The main result of Section 3, Theorem 3.2 motivates a natural question related to uniquenessof inverses between interpolated spaces in abstract setting. Before we formulate a question weintroduce a key definition.A family { F θ } θ ∈ (0 , of interpolation functors is said to be stable if for any Banach couples ~A = ( A , A ) and ~B = ( B , B ) and for every operator S : ~A → ~B such that the restriction S θ ∗ of S to F θ ∗ ( ~A ) is invertible for some θ ∗ ∈ (0 , 1) there exists ε > θ ∈ I ( θ ∗ ) = ( θ ∗ − ε, θ ∗ + ε ), we have(i) S θ : F θ ( ~A ) → F θ ( ~B ) are invertible operators;(ii) S − θ : F θ ( ~B ) → F θ ( ~A ) agrees with S − θ ∗ : F θ ∗ ( ~B ) → F θ ∗ ( ~A ) on B ∩ B , i.e., S − θ y = S − θ ∗ y for all y ∈ B ∩ B ;(iii) sup θ ∈ I ( θ ∗ ) || S − θ || F θ ( ~B ) → F θ ( ~A ) ≤ C || S − θ ∗ || F θ ∗ ( ~B ) → F θ ∗ ( ~A ) for some C = C ( θ ∗ ).An immediate consequence of Theorem 3.2 is the following. Corollary 4.1. If ~ X = ( X , X ) is a Banach couple of translation and rotation invariantpseudolattices, then the following family of interpolation functors { F θ } θ ∈ (0 , is stable, where F θ ( A , A ) ∼ = ( A , A ) ~χ,e θ for any Banach couple ( A , A ) . { F θ } θ ∈ (0 , be a stable family of interpolation functors and T : ( X , X ) → ( Y , Y ) bea bounded linear operator from a Banach couple ~X = ( X , X ) to a Banach couple ~Y = ( Y , Y ).Then the set of all θ ∈ (0 , 1) for which T : F θ ( X , X )) → F θ ( Y , Y ) is invertible, is open, soit is a union of open disjoint intervals. These intervals we will call intervals of invertibility of T with respect to the family { F θ } θ ∈ (0 , .Let I ⊂ (0 , 1) be any interval of invertibility of T . In this section we are interested inthe following question: is it true that for any θ , θ ′ ∈ I the inverses T − θ and T − θ ′ agree on F θ ( Y , Y ) ∩ F θ ′ ( Y , Y )? We point out that this problem is very important for PDEs (see, forexample, discussions in [13]).We will often use the following simple proposition. Proposition 4.2. Let ~A = ( A , A ) and ~B = ( B , B ) be Banach couples and let T : ~A → ~B be an invertible operator. Then, the following conditions are equivalent :(i) ( T | A ) − y = ( T | A ) − y, for all y ∈ B ∩ B ;(ii) T : A + A → B + B is invertible ;(iii) For any interpolation functor G an operator T | G ( ~A ) : G ( ~A ) → G ( ~B ) is invertible.Proof. (i) ⇒ (ii). Since T : ~A → ~B is invertible hence T : A + A → B + B is surjective andtherefore it is enough to prove that T : A + A → B + B is injective. Let x ∈ A + A and T x = 0. Then there exists a decomposition x = x + x , x ∈ A , x ∈ A . From T x = 0 itfollows that y = T x = − T x ∈ B ∩ B . Then from (i), we get that x = − x , and whence x = 0.(ii) ⇒ (iii). Let denote by T − the inverse operator to T : A + A → B + B . Clearly T − is a bounded linear operator from ( B , B ) to ( A , A ) and so T − | G ( ~B ) : G ( ~B ) → G ( ~A ) is aninverse operator to T | G ( ~A ) : G ( ~A ) → G ( ~B ).The same arguments show that (ii) ⇒ (i). Since G ( ~A ) = A + A is an interpolation functor,the implication (iii) ⇒ (ii) follows.Now we are ready to state and prove the following result. Theorem 4.3. Let T : ( X , X ) → ( Y , Y ) be a linear bounded operator and I ⊂ (0 , be an in-terval of invertibility of T with respect to the stable family of interpolation functors { F θ } θ ∈ (0 , .If Y ∩ Y is dense in F α ( ~Y ) ∩ F β ( ~Y ) for all α , β ∈ (0 , , then for any θ , θ ∈ I the inverseoperators T − θ and T − θ agree on F θ ( ~Y ) ∩ F θ ( ~Y ) .Proof. Since Y ∩ Y is dense in F θ ( ~Y ) ∩ F θ ( ~Y ), for any y ∈ F θ ( ~Y ) ∩ F θ ( ~Y ) there is a sequence { y n } ⊂ Y ∩ Y which converges to y in F θ ( ~Y ) and F θ ( ~Y ). Our hypothesis that the familyof functors { F θ } θ ∈ (0 , is stable implies that T − θ y n = T − θ y n . Clearly that x n := T − θ y n = T − θ y n → T − θ y in F θ ( ~X ). We also have that x n → T − θ y in F θ ( ~X ). In consequence thesequence ( x n ) converges to elements T − θ y and T − θ y in X + X . Thus T − θ y = T − θ y asrequired. Remark 4.4. The condition that Y ∩ Y is dense in F α ( ~Y ) ∩ F β ( ~Y ) for all α, β ∈ (0 , 1) israther restrictive. For example, it is not true for F θ ( Y , Y ) = ( Y , Y ) θ, ∞ for any non-trivial15anach couple ( Y , Y ), i.e., such that Y ∩ Y is not closed in Y + Y . However if { F θ } θ ∈ (0 , isa family of regular K -functors, then from Remark 3.6.5 in [4] easily follows that this conditionis fulfilled. In particular it is true for families of functors given by F θ ( Y , Y ) = ( Y , Y ) ◦ θ, ∞ and F θ ( Y , Y ) = ( Y , Y ) θ,q with q < ∞ .In the next proposition we show that under approximation hypothesis on ( Y , Y ) the densitycondition required in Theorem 4.3 holds. Let us remind that the functor F θ is said to be of type θ if for any Banach couple ~A = ( A , A ), we have continuous inclusions ~A θ, ֒ → F θ ( ~A ) ֒ → ~A θ, ∞ . Proposition 4.5. Assume that a Banach couple ~Y = ( Y , Y ) satisfies the following approxi-mation condition : there exists a sequence { P n } ∞ n =1 of linear operators P n : Y + Y → Y ∩ Y such that sup n ≥ k P n k ~Y → ~Y < ∞ and k P n y − y k Y → as n → ∞ . Then, for any pair of regularinterpolation functors F θ and F θ of type θ and θ , respectively, we have that Y ∩ Y is densein F θ ( ~Y ) ∩ F θ ( ~Y ) .Proof. At first we note that there exists a constant C > j ∈ { , } , wehave k y k F θj ≤ C k y k − θ j Y k y k θ j Y , y ∈ Y ∩ Y . Hence, we get that for all y ∈ Y ∩ Y and each j ∈ { , } ,lim n →∞ k P n ( y ) − y k F θj ( ~Y ) = 0 . By interpolation property, it follows that sup n ≥ k P n k F θj ( ~Y ) → F θj ( ~Y ) < ∞ . Since the functors areregular, we deduce that lim n →∞ k P n ( y ) − y k F θ ( ~Y ) ∩ F θ ( ~Y ) = 0for every y ∈ F θ ( ~Y ) ∩ F θ ( ~Y ), as required.We note that Lions [17] showed that a very wide class of Banach couples satisfy the approxi-mation condition used in the above proposition.We will say that a family of interpolation functors { F θ } θ ∈ (0 , satisfies the global (∆)-condition if for any Banach couple ~A = ( A , A ) and for any θ , θ with 0 < θ < θ < 1, wehave continuous inclusions F θ ( ~A ) ∩ F θ ( ~A ) ֒ → \ θ <θ<θ F θ ( ~A ) ֒ → ( F θ ( ~A )) c ∩ ( F θ ( ~A )) c , (4.1)where the norm in T θ <θ<θ F θ ( ~A ) is given by k a k T θ <θ<θ F θ ( ~A ) = sup θ <θ<θ k a k F θ ( ~A ) . (4.2)and the Gagliardo completion ( F θ i ( ~A )) c , j ∈ { , } is taken with respect to the sum F θ ( ~A ) + F θ ( ~A ).In what follows we will use the following obvious observation.16 roposition 4.6. Let { F θ } θ ∈ (0 , and { G θ } θ ∈ (0 , be families of interpolation functors. Sup-pose that there exist positive functions C , C defined on (0 , which are bounded on everycompact subinterval of (0 , and such that F θ ( ~X ) = G θ ( ~X ) with C ( θ ) k · k G θ ( ~X ) ≤ k · k F θ ( ~X ) ≤ C ( θ ) k · k G θ ( ~X ) for every Banach couple ~X and all θ ∈ (0 , . Then (i) The family { F θ } θ ∈ (0 , is stable if and only if the family { G θ } θ ∈ (0 , is stable. (ii) The family { F θ } θ ∈ (0 , satisfies the global (∆) -condition if, and only if, the family { G θ } θ ∈ (0 , satisfies the global (∆) -condition. To state and prove the theorem on stability of inverses on interpolated spaces we needone more definition. We say that a family of interpolation functors { F θ } θ ∈ (0 , satisfies thereiteration condition if for any Banach couple ~A = ( A , A ) and for any θ , θ , λ ∈ (0 , 1) wehave F λ ( F θ ( ~A ) , F θ ( ~A )) = F (1 − λ ) θ + λθ ( ~A ) . Theorem 4.7. Let T : ( X , X ) → ( Y , Y ) be a linear bounded operator and I ⊂ (0 , be an in-terval of invertibility of T with respect to the stable family of interpolation functors { F θ } θ ∈ (0 , .If { F θ } θ ∈ (0 , satisfies the global (∆) –and reiteration conditions, then for any θ , θ ∈ I , theinverse operators T − θ and T − θ agree on F θ ( ~Y ) ∩ F θ ( ~Y ) .Proof. We deduce from Proposition 4.2 that it is enough to prove invertibility of the operator T : F θ ( ~X ) + F θ ( ~X ) → F θ ( ~Y ) + F θ ( ~Y )Since T : F θ j ( X , X ) → F θ j ( Y , Y ) for j ∈ { , } , is invertible, T : F θ ( X , X )+ F θ ( X , X ) → F θ ( Y , Y ) + F θ ( Y , Y ) is surjective. Then, it is enough to prove that T : F θ ( X , X ) + F θ ( X , X ) → F θ ( Y , Y ) + F θ ( Y , Y ) is injective. Let x ∈ F θ ( X , X ) + F θ ( X , X ) and T x = 0. Then x = x + x , where x j ∈ F θ j ( X , X ) , ∈ { , } . From T x = 0, we have y = T x = − T x ∈ F θ ( Y , Y ) ∩ F θ ( Y , Y ) . Since { F θ } θ ∈ (0 , satisfies the global (∆)-condition, y ∈ F θ ( ~Y ) for all θ ∈ ( θ , θ ) and k y k T θ <θ<θ F θ ( ~Y ≤ C k y k F θ ( ~Y ) ∩ F θ ( ~Y ) . ( ∗ )Fix θ , θ ∈ I . Then for any θ ∈ ( θ , θ ) the operator T θ = T | F θ ( ~X ) is invertible and so x θ := T − θ y is well defined.We claim that x θ does not depend on θ ∈ ( θ , θ ). To see this let us consider the couples( e X , e X ) = ( F θ ( ~X ) , F θ ( ~X )) , ( e Y , e Y ) = ( F θ ( ~Y ) , F θ ( ~Y )) . Let e T λ be the restriction of the operator T : ( X , X ) → ( Y , Y ) to F λ ( e X , e X ). Since θ =(1 − λ ) θ + λθ for some λ ∈ (0 , e T λ x = T θ x , for any x ∈ F λ ( e X , ˜ X ) = F θ ( X , X ). Hence˜ T λ : F λ ( e X , e X ) → F λ ( e Y , e Y )17s invertible for all λ ∈ (0 , λ , λ ∈ (0 , F θ combined with the compactness of thesubinterval [ λ , λ ] of (0 , 1) yields that inverse operators e T − λ : F λ ( e Y , e Y ) → F λ ( e X , e X ) and e T − λ : F λ ( ˜ Y , e Y ) → F λ ( ˜ X , e X )agree on e Y ∩ e Y = F θ ( Y , Y ) ∩ F θ ( Y , Y ). Hence the element x θ = T − θ y = e T − λ y is in-dependent of θ ∈ ( θ , θ ) and we denote it by ¯ x . Moreover, the element ¯ x belongs to the set T θ <θ<θ F θ ( ~X ) and k ¯ x k T θ <θ<θ F θ ( ~X ) < ∞ . Indeed, from invertibility of the operator T on thewhole interval I , stability of the family F θ and compactness of the interval [ θ , θ ], we get thatsup θ <θ<θ k T − θ k F θ ( Y ,Y ) → F θ ( X ,X ) < ∞ . Hence from the shown above estimate ( ∗ ), we obtain k ¯ x k T θ <θ<θ F θ ( ~X ) ≤ sup θ <θ<θ k T − θ k F θ ( ~Y ) → F θ ( ~X ) k y k F θ ( ~Y ) ∩ F θ ( ~Y ) < ∞ . Thus using the right hand continuous inclusion in the definition of the global (∆)-condition,we conclude that ¯ x ∈ ( F θ ( ~X )) c ∩ ( F θ ( ~X )) c . Now to finish the proof we decompose the element x as x = x + x = ( x − ¯ x ) + ( x + ¯ x ) . Since T x = − T x = T ¯ x , it is clear that x − ¯ x ∈ ( F θ ( ~X )) c ∩ ker T and x + ¯ x ∈ ( F θ ( ~X )) c ∩ ker T. Invertibility of the operator T on F θ j ( ~X ) implies injectivity of T on ( F θ j ( ~X )) c for each j ∈ { , } .This implies that both x − ¯ x and x + ¯ x are equal to zero. Consequently x = 0 and so theoperator T : F θ ( ~X ) + F θ ( ~X ) → F θ ( ~Y ) + F θ ( ~Y ) is invertible.To show applications to complex and real interpolation methods of the above results weneed a lemma. Lemma 4.8. The families { [ · ] θ } θ ∈ (0 , of the Calde´on functors, as well as { ( · ) θ,q } θ ∈ (0 , with ≤ q ≤ ∞ of the Lions–Peetre interpolation functors, are both stable.Proof. Let { G θ } := { [ · ] θ } θ ∈ (0 , . At first we note that it is shown in [11] that for any Banachcouple ( A , A ) we have ( A , A ) θ,e θ ∼ = [ A , A ] λθ , θ ∈ (0 , , where [ A , A ] λθ is the ”periodic” interpolation space with λ = 2 π . It follows immediately fromthe definition of the periodic interpolation space that[ A , A ] πθ ֒ → [ A , A ] θ with norm of the inclusion map less or equal than 1. Analysis of the proof of Equivalence in[7, p. 1008] shows that [ A , A ] θ ֒ → [ A , A ] πθ , θ ∈ (0 , C ( θ ). Standard calculus shows that thereexists a positive constant K independent of θ such that C ( θ ) ≤ Kθ (1 − θ ) . Altogether yields that the family { F θ } := { ( · ) ( F C,F C ) ,e θ } θ ∈ (0 , satisfies F θ ( A , A ) = G θ ( A , A ) , θ ∈ (0 , , where the constants of equivalence of norms are bounded on any compact subinterval of (0 , { G θ } := { ( · ) θ,q } θ ∈ (0 , for any fixed 1 ≤ q ≤ ∞ . Put { F θ } := { ( · ) ( ℓ q ,ℓ q ) ,e θ } θ ∈ (0 , (cid:9) . It was noticed in Section 2 that F θ ( ~A ) ∼ = G θ ( ~A )for all Banach couples. It is well known that F θ ( ~A ) = G θ ( ~A ) up to equivalence of norms.Standard calculus shows there exist absolute positive constants C > C > 0, independenton θ and q ∈ [1 , ∞ ), such that C k · k F θ ( ~A ) ≤ k · k G θ ( ~A ) ≤ Bθ (1 − θ ) k · k G θ ( ~A ) . Again applying Corollary 4.1 and Proposition 4.6 we are done.We are ready to prove the compatibility theorem for the family of real interpolation functors. Theorem 4.9. Let ≤ q ≤ ∞ and let T : ( X , X ) → ( Y , Y ) be a linear bounded operatorand I ⊂ (0 , be an interval of invertibility of T with respect to the family { ( · ) θ,q } θ ∈ (0 , ofreal interpolation functors. Then for any θ , θ ∈ I the inverse operators T − θ and T − θ agreeon ( Y , Y ) θ ,q ∩ ( Y , Y ) θ ,q .Proof. It is well known that the family of real interpolation functors satisfies the reiterationcondition. Moreover stability of this family follows from Lemma 4.8. Thus in order to applyTheorem 4.7, we only need to check that this family satisfies the global (∆)-condition (4.1).Since (( A , A ) θ,q ) c ∼ = ( A , A ) θ,q ), it is enough to prove that for any Banach couple ~A =( A , A ), we have ( A , A ) θ ,q ∩ ( A , A ) θ ,q = \ θ <θ<θ ( A , A ) θ,q . Let x ∈ ( A , A ) θ ,q ∩ ( A , A ) θ ,q and θ < θ < θ . Then k x k θ,q = (cid:18) ∞ ˆ (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q ≤ (cid:18) ∞ ˆ (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q + (cid:18) ∞ ˆ (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q ≤ {k x k θ ,q , k x k θ ,q } . This yields ( A , A ) θ ,q ∩ ( A , A ) θ ,q ֒ → T θ <θ<θ ( A , A ) θ,q . x ∈ T θ <θ<θ ( A , A ) θ,q and 0 < a < b < ∞ . Since θ ∈ ( θ , θ ) is arbitrary, (cid:18) b ˆ a (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q ≤ b ( θ − θ ) (cid:18) b ˆ a (cid:0) t − θ K ( t, x ; ~A )) q dtt (cid:19) /q . Similarly, we get that (cid:18) b ˆ a (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q ≤ a ( θ − θ ) (cid:18) b ˆ a (cid:0) t − θ K ( t, x ; ~A ) (cid:1) q dtt (cid:19) /q . Taking in account that these inequalities are correct for any θ ∈ ( θ , θ ) and for arbitrary a and b , we get that max {k x k θ ,q , k x k θ ,q } ≤ k x k T θ <θ<θ ~A θ,q . Hence T θ <θ<θ ( A , A ) θ,q ֒ → ( A , A ) θ ,q ∩ ( A , A ) θ ,q . Similarly, we prove the case p = ∞ .From Theorem 4.7 also follows the compatibility theorem for the family { F θ } θ ∈ (0 , of com-plex interpolation functors: F θ ( A , A ) = [ A , A ] θ . (4.3) Theorem 4.10. Let T : ( X , X ) → ( Y , Y ) be an operator between couples of complex Ba-nach spaces and let I ⊂ (0 , be an interval of invertibility of T with respect to the family ofinterpolation functors defined by ( ) . Then for any θ , θ ∈ I the inverse operators T − θ and T − θ agree on [ Y , Y ] θ ∩ [ Y , Y ] θ . Proof. As well as in the proof of Theorem 4.9 it is enough to prove the global (∆)-conditionfor the family { [ · ]) θ } for arbitrary Banach couple ( A , A ):[ A , A ] θ ∩ [ A , A ] θ ֒ → \ θ <θ<θ [ A , A ] θ ֒ → ([ A , A ] θ ) c ∩ ([ A , A ] θ ) c , where Gagliardo completion ([ A , A ] θ j ) c for j ∈ { , } is taken with respect to the sum[ A , A ] θ + [ A , A ] θ . Since the reiteration formula[[ A , A ] θ , [ A , A ] θ ] λ = [ A , A ] (1 − λ ) θ + λθ holds with equality of norms for any θ , θ , λ ∈ (0 , 1) (see [7]). Hence, for any x ∈ [ A , A ] θ ∩ [ A , A ] θ , we have k x k (1 − λ ) θ + λθ ≤ k x k − λθ k x k λθ ≤ max {k x k θ , k x k θ } . This proves that [ A , A ] θ ∩ [ A , A ] θ ֒ → \ θ <θ<θ [ A , A ] θ . The proof of Theorem 4.7.1 in [3] shows that for any x ∈ [ A , A ] θ k x k θ, ∞ ≤ k x k θ . k x k cA = sup t> K ( t, x ; ~A ) and k x k cA = sup t> K ( t, x ; ~A ) t , we get that sup θ <θ<θ k x k θ = sup <λ< k x k [ ~A θ ,θ ] λ ≥ sup <λ< k x k ( ~A θ , ~A θ ) λ , ∞ ≥ max {k x k ([ A ,A ] θ ) c , k x k ([ A ,A ] θ ) c } , where the Gagliardo completion ([ A , A ] θ j ) c , j ∈ { , } is taken with respect to the sum[ A , A ] θ + [ A , A ] θ . Thus we conclude that the second required continuous inclusion \ θ <θ<θ [ A , A ] θ ֒ → ([ A , A ] θ ) c ∩ ([ A , A ] θ ) c holds and so this completes the proof. Theorem 4.11. Let T : ( X , X ) → ( Y , Y ) be an operator between couples of complex Banachspaces. If T : [ X , X ] θ ∗ → [ Y , Y ] θ ∗ is invertible for some θ ∗ ∈ (0 , , then T : ( X , X ) θ ∗ ,q → ( Y , Y ) θ ∗ ,q is invertible for all q ∈ [1 , ∞ ] .Proof. Let θ ∗ ∈ I , where I is an interval of invertibility of T with respect to the family offunctors of complex interpolation. Then there exists ε > θ = θ − ε, θ = θ + ε ∈ I. From Theorem 4.10 follows that inverse operators T − θ and T − θ agree on [ Y , Y ] θ ∩ [ Y , Y ] θ . So from Proposition 4.2 (iii) we obtain invertibility of the operator T : ([ X , X ] θ , [ X , X ] θ ) ,q → ([ Y , Y ] θ , [ Y , Y ] θ ) ,q . To complete the proof it remains to note that([ X , X ] θ , [ X , X ] θ ) ,q = ( X , X ) θ ∗ ,q and ([ Y , Y ] θ , [ Y , Y ] θ ) ,q = ( Y , Y ) θ ∗ ,q . We conclude with the following result about the connections between spectrum of interpo-lated operators. The result is an immediate consequence of Theorem 4.11. Theorem 4.12. Let ~ X = ( X , X ) be a Banach couple of translation and rotation invariantpseudolattices and let the family { F θ } := { ( · ) ~ X ,e θ } θ ∈ (0 , be such the reiteration condition holdsfor a complex Banach couple ( X , X ) . If { F θ } satisfies a global (∆) -condition for ( X , X ) then, for any operator T : ~X → ~X and all q ∈ [1 , ∞ ] , we have σ (cid:0) T, ~X θ,p (cid:1) ⊂ σ (cid:0) T, F θ ( ~X ) (cid:1) . As a consequence, we obtain the following corollary. Corollary 4.13. Let ( X , X ) be a couple of complex Banach spaces. Then, for any operator T : ( X , X ) → ( X , X ) and for all q ∈ [1 , ∞ ] , we have σ ( T, ~X θ,q ) ⊂ σ ( T, [ ~X ] θ ) . 21e conclude with the following remark that Albrecht and M¨uller gave an example of a Ba-nach couple ~X and and operator : ~X → ~X for which σ ( T, ~X θ, ) = σ ( T, [ ~X ] θ ) (see [1, Example12]). Throughout this section (Ω , Σ , µ ) denotes a σ -finite measure space. The symbol L ( µ ) := L (Ω , Σ , µ ) stands for the space of (equivalence classes of µ -a.e. equal) real-valued measurablefunctions on Ω with the topology of convergence in measure on µ -finite sets. As usual the order | f | ≤ | g | means that | f ( t ) | ≤ | g ( t ) | for µ -almost all t ∈ Ω.If a Banach space X ⊂ L ( µ ) contains an element which is strictly positive µ -a.e. on Ω and X is solid (meaning that f ∈ X with k f k X ≤ k g k X whenever | f | ≤ | g | with f ∈ L ( µ ) and g ∈ X ), then X is said to be a Banach lattice on (Ω , Σ , µ )). A Banach lattice X is said to havethe Fatou property , if for any sequence { f n } of non-negative elements from X such that f n ↑ f for f ∈ L (Ω) and sup (cid:8) k f n k X ; n ∈ N (cid:9) < ∞ , one has f ∈ X and k f n k X ↑ k f k X .Let X and Y be Banach lattices. A linear operator T : X → Y is said to be positive (resp., homomorphism) if T x ≥ x ≥ T x ∧ T y = 0 whenever x ∧ y = 0).A homomorphism which is additionally a bijection is called an order isomorphism . It is wellknown that a linear bijection T : X → Y is an order isomorphism if and only if T and T − areboth positive (see [2, Theorem 7.3]).This section elaborates on an unpublished result of Milman [20] on a strong variant ofShnieberg result that states that, under some mild conditions, and in the context of Banachlattices, invertibility of a bounded positive operator at one point of the scale of Calder´on spacefor Banach function lattices implies invertibility at all points in the interior scale. Combiningwith our previous results we obtain a variant of this result for the classical real interpolationspaces between Banach lattices.We recall that the Calder´on product X − θ X θ defined for any couple ( X , X ) of Banachlattices on a measure space (Ω , Σ , µ ) consists of all f ∈ L ( µ ) such that | f | ≤ λ | f | − θ | f | θ µ -a.e. for some λ > f j ∈ X j with k f j k X j ≤ j ∈ { , } . It is well known (see [5]) that X − θ X θ is a Banach lattice endowed with the norm k f k = inf (cid:8) λ > | f | ≤ λ | f | − θ | f | θ , k f k X ≤ , k f k X ≤ (cid:9) . In what follows for simplicity of notation, we also write for short X θ instead of X − θ X θ .For the reader’s convenience, we include the proof of the mentioned above result. Theorem 5.1. Let T : ( X , X ) → ( Y , Y ) be a positive operator between couples of Banachlattices with the Fatou property. Assume that T : X − θ X θ → Y − θ Y θ is an order isomor-phism for some θ ∈ (0 , . Then T : X − θ X θ → Y − θ Y θ is an order isomorphism for all θ ∈ (0 , .Proof. Notice that for any couple ( E , E ! ) of Banach lattices with the Fatou property and forevery θ ∈ (0 , E − θ E θ is a Banach lattice with the Fatou property (see [19]). Thus by use ofextrapolation formula of Cwikel–Nilsson [9, Theorem 3.5], we have k f k E − θ E θ = sup n(cid:13)(cid:13) | g | − α | f | α (cid:13)(cid:13) /αE − α ( E − θ E θ ) α ; k g k E ≤ o , α, θ ∈ (0 , . T : X − θ X θ → Y − θ Y θ , θ ∈ (0 , . Suppose that for some C > f ≥ X θ we have C k f k X θ ≤ k T f k Y θ . ( ∗ )We will use the following easily verified reiteration formula true for an arbitrary couple ofBanach lattices, which is true for all α , θ and θ in [0 , X − θ X θ ) − α ( X − θ X θ ) α = X − β X β , where β = (1 − α ) θ + αθ . We also require the following property of any positive operator P : X → Y between Banach lattices which says that if 0 ≤ x, y ∈ X and θ ∈ (0 , P ( x − θ y θ ) ≤ P ( x ) − θ P ( y ) θ . We may assume without loss of generality that k T k X j → Y j ≤ j ∈ { , } and also that0 < θ < θ < 1. Thus, we can find α ∈ (0 , 1) such that θ = αθ . Suppose f ∈ X θ isnonnegative. Combining Cwikel–Nilsson formula shown above with the mentioned property ofpositive operators and our hypothesis we obtain k T f k Y θ = sup n(cid:13)(cid:13) | g | − α ( T f ) α (cid:13)(cid:13) /αY θ ; k g k Y ≤ o ≥ sup n(cid:13)(cid:13) ( T | x | ) − α ( T f ) α (cid:13)(cid:13) /αY θ ; k x k X ≤ o ≥ sup n(cid:13)(cid:13) T ( | x | − α f α ) (cid:13)(cid:13) /αY θ ; k x k X ≤ o . In consequence, our hypothesis ( ∗ ) on T and the mentioned extrapolation formula yield therequired estimate (cid:13)(cid:13) T f (cid:13)(cid:13) Y θ ≥ C /α sup n(cid:13)(cid:13) | x | − α f α (cid:13)(cid:13) /αY θ ; k x k X ≤ o = C /α (cid:13)(cid:13) f (cid:13)(cid:13) X θ and this completes the proof.In the sequel when the complex methods are applied to a couple ( X , X ) of Banach lattices,we mean that X j := X j ( C ) is a complexification of X j for j ∈ { , } . If X is an intermediateBanach space with respect to a couple ~X = ( X , X ), we let X ◦ be the closed hull of X ∩ X in X .We conclude with the following result. Theorem 5.2. Let ~X = ( X , X ) and ~Y = ( Y , Y ) be couples of regular Banach lattices withthe Fatou property and let T : X + X → Y + Y be a positive operator. If T : X − θ ∗ X − θ ∗ → Y − θ ∗ Y θ ∗ is an order isomorphism for some θ ∗ ∈ (0 , , then T : X − θ X θ → Y − θ Y θ , T : ( X , X ) θ,p → ( Y , Y ) θ,p are order isomorphisms for all θ ∈ (0 , , p ∈ [1 , ∞ ] .Proof. Since the couples are regular, we have that X ∩ X is dense in X α and Y ∩ Y is densein Y α for all α ∈ (0 , h X , X i θ ∗ = (cid:0) X − θ ∗ X θ ∗ (cid:1) ◦ = X − θ ∗ X θ ∗ , h Y , Y i θ ∗ = Y − θ Y θ ∗ . Thus, by Theorem 5.1, we deduce that T : h X , X i θ → h Y , Y i θ is an order isomorphism for all θ ∈ (0 , 1) and so Theorem 4.3 applies. References [1] E. Albrecht and V. M¨uller, Spectrum of interpolated operators , Proc. Amer. Math. Soc. 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Verchota, The Dirichlet problem in L p for the biharmonic equation onLipschitz domains , Amer. J. Math. (1992), no. 5, 923-972.[24] T. J. Ransford, The spectrum of an interpolated operator and analytic multivalued func-tions , Pacific J. Math. (1986), no. 2, 445-466.[25] I. Ja Shneiberg, Spectral properties of linear operators in interpolation families of Banachspaces , Math. Issled. (1974), 214–229 (Russian).[26] M. Zafran, Spectral theory and interpolation of operators , J. Funct. Anal. (1980), no. 2,185-204.Department of Mathematics (MAI)Link¨oping University, SwedenE-mail: [email protected] Department of Mathematics (MAI)Link¨oping University, SwedenE-mail: [email protected] Faculty of Mathematics and Computer ScienceAdam Mickiewicz University in Pozna´nUniwersytetu Pozna´nskiego 461-614 Pozna´n, PolandE-mail: mastylo @ math.amu.edu.plmath.amu.edu.pl