Closed operator functional calculus in Banach modules and applications
aa r X i v : . [ m a t h . F A ] J u l Closed operator functional calculusin Banach modules and applications
Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. Uskova
Abstract.
We describe a closed operator functional calculus in Banachmodules over the group algebra L ( R ) and illustrate its usefulness witha few applications. In particular, we deduce a spectral mapping theo-rem for operators in the functional calculus, which generalizes some ofthe known results. We also obtain an estimate for the spectrum of aperturbed differential operator in a certain class. Keywords.
Functional Calculus, Banach modules, Asymptotic spectralanalysis, Spectral mapping theorem.
1. Introduction
The goal of this paper is to describe a closed operator functional calculus inBanach modules over the group algebra L ( R ) and to illustrate its usefulnesswith a few applications. The functional calculus was introduced in [9] in orderto obtain several non-commutative extensions of Wiener’s 1 /f lemma [27].The first application discussed in this paper (see Theorems 3.5 and 3.12)gives several versions of the spectral mapping theorem [3, 5, 16, 24].The second application of the functional calculus is an estimate of thespectrum σ ( L ) of a differential operator L = A − V : D ( A ) ⊆ L ( R ) → L ( R ), A = − i ddt . The domain D ( A ) is chosen to be the Sobolev space W , ( R ) of absolutely continuous functions with the (almost everywhere)derivative in L ( R ), and the operator V : D ( A ) ⊆ L ( R ) → L ( R ) is chosento be of the form ( V x )( t ) = v ( t ) x ( − t ) , t ∈ R , v ∈ L ( R ) . (1.1)In [10, 11], we performed the spectral analysis for an analogous operatoron L ([0 , ω ]). In fact, in that case, σ ( L ) is discrete and differs from σ ( A ) byan ℓ sequence. Here, σ ( A ) = R , and we end up estimating a region in C that contains σ ( L ). We cite [10, 11, 14, 21, 26] and references therein for the Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. Uskovamotivation of studying differential operators with an involution, such as thereflexion operator V .The resulting estimate is contained in the following theorem. Theorem 1.1.
Consider the operator L = − i ddt − V : W , ( R ) ⊆ L ( R ) → L ( R ) with V of the form (1.1) . Then there exists a continuous real-valuedfunction f ∈ L ( R ) such that for any λ ∈ σ ( L ) one has |ℑ mλ | ≤ f ( ℜ eλ ) . Thus, in Theorem 1.1, the spectrum σ ( L ) lies between the graphs ofthe functions f and − f .The remainder of the paper is organized as follows. In Section 2, weintroduce the necessary notions and notation and describe the functionalcalculus. In Section 3, we formulate and prove a few novel versions of thespectral mapping theorem. In Section 4, we prove Theorem 1.1. Finally, Sec-tion 5 contains proofs of a few auxiliary results. F L loc functional calculus. In our exposition of the closed operator functional calculus for generators ofBanach L ( R )-modules, we follow [9]. Let us introduce some notation.We denote by X a complex Banach space and by B ( X ) the Banachalgebra of all bounded linear operators in X . We also assume that X isendowed with a non-degenerate Banach module structure over the groupalgebra L ( R ). The multiplication in L ( R ) is the convolution( f ∗ g )( t ) = Z R f ( s ) g ( t − s ) ds, f, g ∈ L ( R ) , t ∈ R . Definition 2.1.
A complex Banach space X is a Banach module over L ( R ) ifthere is a bilinear map ( f, x ) f x : L ( R ) × X → X which has the followingproperties:1. ( f ∗ g ) x = f ( gx ), f, g ∈ L ( R ), x ∈ X ;2. k f x k ≤ k f k k x k , f ∈ L ( R ), x ∈ X .As usual (see [7, 8] and references therein), by non-degeneracy of themodule we mean that f x = 0 for all f ∈ L ( R ) implies that x = 0. We onlyconsider Banach module structures that are associated with an isometricrepresentation T : R → B ( X ), that is we have T ( t )( f x ) = f t x = f ( T ( t ) x ) , t ∈ R , f ∈ L ( R ) , x ∈ X , (2.1)where f t ( s ) = f ( t + s ), t, s ∈ R . With a slight abuse of notation [8], given f ∈ L ( R ), we shall denote by T ( f ) the operator in B ( X ) defined by T ( f ) x = f x , x ∈ X . Observe that we have kT ( f ) k ≤ k f k , f ∈ L ( R ), by Property2 in Definition 2.1. For the Banach module X , we will also use the notation( X , T ) if we want to emphasize that the module structure is associated withthe representation T .We use the Fourier transform of the form( F ( f ))( ξ ) = b f ( ξ ) = Z R f ( t ) e − itξ dt, f ∈ L ( R ) , losed Operator Functional Calculus 3so that k b f k = √ π k f k , f ∈ L ( R ). We shall denote by F L = F L ( R )the Fourier algebra F ( L ( R )). The inverse Fourier transform of a function h ∈ F L ( R ) will be denoted by ˇ h or F − ( h ). Definition 2.2.
Let ( X , T ) be a non-degenerate Banach L ( R )-module, and N be a subset of X . The Beurling spectrum Λ( N ) = Λ( N, T ) is defined byΛ( N, T ) = { λ ∈ R : f x = 0 for all x ∈ N implies b f ( λ ) = 0 , f ∈ L } . To simplify the notation we shall write Λ( x ) instead of Λ( { x } ), x ∈ X .We refer to [8, Lemma 3.3] for the basic properties of the Beurling spectrum.We also define X comp = { x ∈ X : Λ( x ) is compact } , X Φ = {T ( f ) x : f ∈ L ( R ) , x ∈ X } and X c = { x ∈ X : the function t
7→ T ( t ) x : R → X is continuous } , For any z ∈ C \ R , consider the function f z ∈ L ( R ) whose Fouriertransform is the function φ z : R → C defined by φ z ( λ ) = ( λ − z ) − , λ ∈ R .Hilbert’s resolvent identity holds for the operator-valued function R : C \ R → B ( X ) given by R ( z ) = T ( f z ), z ∈ C \ R . Since the L ( R )-module X is non-degenerate, we have T z ∈ C \ R ker R ( z ) = { } . Therefore [6], R is the resolventof some linear operator A : D ( A ) ⊆ X → X . This operator A is called thegenerator of the L ( R )-module X . We remark that if T : R → B ( X ) is astrongly continuous group representation, then i A is its generator.It is not hard to show that the operators T ( f ), f ∈ L ( R ), provide afunctional calculus for the generator A . Via the isomorphism of L ( R ) and F L ( R ), we also get the functional calculus ˇ T ( b f ) = T ( f ), b f = F ( f ) ∈ F L .It is useful to extend this functional calculus to the space F L loc ( R ) = { h : R → C such that h b ϕ ∈ F L ( R ) for any ϕ ∈ L ( R ) with supp b ϕ compact } .Observe that F L ( R ) ⊂ F L loc ( R ). Moreover, F L loc is also an algebra underpointwise multiplication.For h ∈ F L loc ( R ) we define a (closed) operator ˇ T ( h ) = h ⋄ : D ( h ⋄ ) = D ( ˇ T ( h )) ⊆ X → X in the following way. First, let x ∈ X comp andˇ T ( h ) x = h ⋄ x := ( h b ϕ ) ∨ x = T (( h b ϕ ) ∨ ) x, (2.2)where ϕ ∈ L ( R ) is such that supp b ϕ is compact and b ϕ ≡ x ). The vector ˇ T ( h ) x is well defined in this way because it is independentof the choice of ϕ .Next, we extend the definition of ˇ T ( h ) by taking the closure of the justdefined operator on X comp . In other words, if x n ∈ X comp , n ∈ N , x = lim n →∞ x n ,and y = lim n →∞ h ⋄ x n exists, we let ˇ T ( h ) x = h ⋄ x = y . Lemma 2.7 in [9] showsthat ˇ T ( h ) is then a well-defined closed linear operator and we do, indeed,have ˇ T ( b f ) = T ( f ), f ∈ L ( R ). Moreover, applying [9, Proposition 2.8], weget for a given x ∈ D ( ˇ T ( h )) that T ( t )( h ⋄ x ) = h ⋄ ( T ( t ) x )) and T ( f )( h ⋄ x ) = h ⋄ ( T ( f ) x )) = ( b f h ) ⋄ x, (2.3) Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. Uskova t ∈ R , f ∈ L ( R ), h ∈ F L loc ( R ). We note that D ( ˇ T ( h )) ⊆ X c for all h ∈F L loc . We also note the following useful property that is implied by thedefinition of the operators ˇ T ( h ) and [8, Lemma 3.3]:Λ( ˇ T ( h ) x, T ) ⊆ supp h ∩ Λ( x, T ) , h ∈ F L loc , x ∈ X . (2.4)It is not hard to see that the generator A of the module ( X , T ) satisfies A = ˇ T (id) , (2.5)where id ∈ F L loc ( R ) is the identity function id( ξ ) = ξ , ξ ∈ R . Thus, we havean F L loc functional calculus for the generator A , which we will use to provea few spectral mapping results and construct a similarity transform to obtainan estimate for the spectrum of the perturbed differential operators.We will use the following sufficient condition for functions in L tobelong to F L . For completeness, we provide its proof in Section 5. Lemma 2.1.
Assume f ∈ L ( R ) and b f ∈ W , ( R ) . Then f ∈ L ( R ) and k f k ≤ k b f k k b f ′ k . (2.6)We illustrate the above lemma with the following two examples. Example . For a >
0, consider the “trapezoid function” τ a defined by τ a ( ξ ) = , | ξ | ≤ a, a (2 a − | ξ | ) , a < | ξ | < a, , | ξ | ≥ a. Direct computations show that k τ a k = 2 q a , k τ ′ a k = q a , and τ a = b ϕ a ,where ϕ a ( t ) = 2 sin at sin at πat , t ∈ R . From Lemma 2.1 we conclude that k ϕ a k ≤ · − . We remark that [22,Lemma 1.10.1] yields a better estimate: k ϕ a k ≤ π + π ln 3. An even betterestimate, k ϕ a k ≤ √
3, follows from [25, Proposition 5.1.5].
Example . For a >
0, let ω a ( ξ ) = 1 ξ (1 − τ a ( ξ )) = , | ξ | ≤ a, − a − ξ , − a < ξ ≤ − a, a − ξ , a < ξ ≤ a, ξ , | ξ | > a. Then k ω a k = q − a ≤ . / √ a and k ω ′ a k = q a ≤ . / ( a √ a ). It fol-lows that the functions ψ a defined by b ψ a = ω a satisfy k ψ a k ≤ a q (1 − ln 2)so that k ψ a k ≤ . /a. losed Operator Functional Calculus 5We will also need an estimate for k ψ a k ∞ . Observe that for t > π (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ ω a ( ξ ) e itξ dξ (cid:12)(cid:12)(cid:12)(cid:12) = 1 π (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ ω a ( ξ ) sin( tξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ π (cid:18) (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ a sin( tξ ) ξ dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = 1 π (cid:18) (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ at sin ξξ dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ π + 1 . Since ω a is an odd function, it follows that k ψ a k ∞ ≤ π + 1 . (2.7)Now we use the functions from the above two examples in our F L loc functional calculus. In view of (2.3) and (2.5), we get the following crucialrelationship: AT ( ψ a ) x = ˇ T (id) T ( ψ a ) x = ˇ T (id · ω a ) x = ˇ T ( − τ a ) x = x − T ( ϕ a ) x, (2.8)which holds for every x ∈ D ( A ); by ∈ F L loc we denoted the function ( ξ ) = 1, ξ ∈ R . In fact, since D ( ˇ T ( )) = X c and A is a closed operator, weget that (2.8) holds for all x ∈ X c . Moreover, non-degeneracy of the module( X , T ) implies that the operator AT ( ψ a ) extends uniquely to a boundedoperator I − T ( ϕ a ) ∈ B ( X ). To simplify the notation, given λ ∈ C , we maywrite λ − f instead of λ − f for functions and λ − A instead of λI − A foroperators.We note that the family ( ϕ α ) from Example 2.1 has another usefulproperty: it forms a bounded approximate identity. Definition 2.3.
A family of functions ( φ a ) a> is called a bounded approximateidentity or b.a.i. if k ϕ a k ≤ M for all a > a →∞ k ϕ a ∗ f − f k = 0for all f ∈ L ( R ).Following [9], we call a b.a.i. ( φ α ) a cf-b.a.i. if supp b φ a is compact foreach a >
0. Clearly, the family ( ϕ α ) from Example 2.1 is a cf-b.a.i. Anotheruseful cf-b.a.i. is given by the family γ a ( t ) = a π (cid:18) sin( at/ at/ (cid:19) , t ∈ R , a > . (2.9)Observe that b γ a ( ξ ) = (1 − | ξ | /a ) [ − a,a ] ( ξ ) =: △ a ( ξ ) is the so called trianglefunction and, since γ a is non-negative, k γ a k = △ a (0) = 1.We conclude this section by recalling the following result that containsthe celebrated Cohen-Hewitt factorization theorem [15, 20]. Proposition 2.2 ( [8] , Lemma 4.3).
For any b.a.i. ( φ a ) , we have X c = X Φ = X comp = { x ∈ X : x = lim a →∞ φ a x = x } . Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. Uskova
3. Spectral mapping theorems.
We begin this section by recalling a spectral mapping theorem from [5] whichwe endeavor to extend (see also [16, 24]).
Theorem 3.1 ( [5] , Corollary 1.5.3).
Let X be a non-degenerate Banach L ( R ) -module with the structure associated with a representation T . For f ∈ L ( R ) , σ ( T ( f )) = b f (Λ( X , T )) . To prove our extensions we will need the following two lemmas that, inparticular, give a special case of the above result.
Lemma 3.2.
Assume that K = Λ( X , T ) is compact. Then, for h ∈ F L loc , wehave σ ( ˇ T ( h )) ⊆ h ( K ) .Proof. Observe that h ( K ) is automatically compact as a continuous image ofa compact set. Assume λ / ∈ h ( K ). Then the function u given by u ( t ) = λ − t is analytic in a neighborhood of the compact set h ( K ). The Wiener-L´evytheorem [25, Theorem 1.3.1] ensures existence of a function f ∈ L ( R ) suchthat b f ( ξ ) = λ − h ( ξ ) for ξ in a neighborhood of K . For x ∈ X = X comp , we use(2.3) and [8, Lemma 3.3] to obtain( λI − T ( f )) ˇ T ( h ) x = ˇ T ( h )( λI − T ( f )) x = ˇ T ( λ b f − b f h ) x = ˇ T ( ) x = x, so that λ ∈ ρ ( ˇ T ( h )). Thus, σ ( ˇ T ( h )) ⊆ h ( K ). Lemma 3.3.
For h ∈ F L loc ( R ) , we have σ ( ˇ T ( h )) ⊇ h (Λ( X , T )) .Proof. Assume λ = h ( ξ ) ∈ ρ ( ˇ T ( h )) for some ξ ∈ R and let f ∈ L ( R ) besuch that supp b f is compact, ξ ∈ supp b f , and b f (supp h ) ⊂ ρ ( ˇ T ( h )). Let also Y = T ( f ) X and B = ˇ T ( h ) | Y be the restriction of the operator ˇ T ( h ) to thesubmodule Y . Clearly, B ∈ B ( Y ). We claim that ρ ( B ) = C , which wouldimply Y = { } yielding supp b f ∩ Λ( X , T ) = ∅ . To prove the claim, we firstobserve that ρ ( ˇ T ( h )) ⊆ ρ ( B ). Indeed, since ˇ T ( h ) commutes with T ( f ) by(2.3), we have that for λ ∈ ρ ( ˇ T ( h )) the resolvent operator ( λI − ˇ T ( h )) − alsocommutes with T ( f ) ensuring ( λI − B ) − = ( λI − ˇ T ( h )) − | Y . Using (2.4),we get b f (Λ( Y )) ⊆ b f (supp h ) ⊂ ρ ( ˇ T ( h )) ⊆ ρ ( B ) . Secondly, Lemma 3.2 implies b f (Λ( Y )) c ⊆ ρ ( B ), and the claim is established.It follows that supp b f ∩ Λ( X , T ) = ∅ , and hence ξ / ∈ Λ( X , T ). Thus, σ ( ˇ T ( h )) ⊇ h (Λ( X , T )) and, since the spectrum is closed, σ ( ˇ T ( h )) ⊇ h (Λ( X , T )).Thus, to extend Theorem 3.5 to the F L loc setting we only need ananalog of Lemma 3.2 for the case when Λ( X ) is not necessarily compact.This, however, may not always hold at this level of generality as we can nolonger use the Wiener-L´evy theorem. We offer several ways to circumvent theproblem.First, we present a result that is immediate from the proof of Lemma3.2.losed Operator Functional Calculus 7 Proposition 3.4.
Let h ∈ F L loc and λ ∈ C . Assume that there exists a func-tion g λ ∈ F L loc such that g λ = ( λ − h ) − in a neighborhood of Λ( X , T ) and ˇ T ( g λ ) belongs to B ( X ) . Then λ ∈ ρ ( ˇ T ( h )) . This motivates the following definition.
Definition 3.1.
Let X = ( X , T ) be a non-degenerate Banach L ( R )-module.A function h ∈ F L loc ( R ) is called X -regular if for any λ / ∈ h (Λ( X , T ) thereexists a function g λ ∈ F L loc ( R ) such that g λ ( ξ )( λ − h ( ξ )) = 1 , (3.1)for every ξ in a neighborhood of Λ( X , T ), and ˇ T ( g λ ) belongs to B ( X ).Clearly, it would be sufficient for the functions g λ in the above definitionto belong to F L . Hence, by the Wiener-L´evy theorem, if Λ( X ) is compact,every h ∈ F L loc ( R ) is X -regular.The next result is now immediate. Theorem 3.5.
Assume that h ∈ F L loc ( R ) is X -regular. Then σ ( ˇ T ( h )) = h (Λ( X , T )) . Moreover, given λ / ∈ h (Λ( X , T )) , we have ( λ − ˇ T ( h )) − =ˇ T ( g λ ) = T (ˇ g λ ) , where g λ ∈ F L loc ( R ) is defined by (3.1) . The following result shows that Theorem 3.5 does indeed generalizeTheorem 3.1.
Proposition 3.6.
Any function h ∈ F L ( R ) is X -regular for any X .Proof. Observe that by the Riemann-Lebesgue lemma if 0 / ∈ h (Λ( X , T )) thenΛ( X , T ) is compact, and the result follows.Assume now that 0 = λ / ∈ h (Λ( X , T )). Then, without loss of generalitywe may assume that λ / ∈ h ( R ). Indeed, if that was not the case, we would have0 = λ ∈ h ( R ) and h − ( { λ } ) would be a compact set disjoint from Λ( X , T ).We could then find φ ∈ F L with compact support that is disjoint fromΛ( X , T ) and such that λ / ∈ ( h + φ )( R ). We would then apply the followingargument to h + φ instead of h .A modification of the Wiener-L´evy theorem (see [25, Theorem 1.3.4] or[17]) or a special case of the Bochner-Phillips theorem (see [9, Theorem 10.3]or [13]) show that g λ = ( λ − h ) − = λ − + e h for some e h ∈ F L . Then ˇ T ( g λ ) = λ − I + ˇ T ( e h ) ∈ B ( X ), and the resultfollows.Another sufficient condition for X -regularity follows from Lemma 2.1. Proposition 3.7.
Assume that h ∈ F L loc ( R ) is such that for every λ / ∈ h (Λ( X , T )) there exists a function g λ ∈ W , ( R ) that satisfies (3.1) in aneighborhood of h (Λ( X , T )) . Then h is X -regular. In particular, every poly-nomial is X -regular for any X . Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. Uskova
Proof.
The first assertion follows immediately from Lemma 2.1. To prove thesecond one, we note that for every polynomial p and λ / ∈ p ( R ) = p ( R ) thefunction ( λ − p ( · )) − belongs to W , ( R ). This shows that p is X -regularfor any X such that Λ( X , T ) = R . In case Λ( X , T ) = R , given λ ∈ p ( R ) \ p (Λ( X , T )), we observe that p − ( { λ } ) is a finite set that does not intersectΛ( X , T ). Hence, there is an infinitely many times differentiable function φ λ ∈F L with compact support that is disjoint from Λ( X , T ) and such that λ / ∈ ( p + φ λ )( R ). Then g λ = ( λ − p − φ λ ) − ∈ W , ( R ), and the result follows.The following well-known result now follows immediately from (2.5) andthe fact that the Beurling spectrum is a closed set. Corollary 3.8.
The generator A of a non-degenerate Banach L ( R ) -module X satisfies σ ( A ) = Λ( X ) . In the context of Proposition 3.7, Lemma 2.1 also allows us to estimatethe resolvent of the operators ˇ T ( h ). Corollary 3.9.
Assume that h ∈ F L loc ( R ) , λ / ∈ h ( R ) , and the function g λ defined by (3.1) belongs to W , ( R ) . Then (cid:13)(cid:13) ( λ − ˇ T ( h )) − (cid:13)(cid:13) ≤ q k g λ k k g ′ λ k . The following definition allows us to provide yet another example of X -regularity. Definition 3.2.
We say that h ∈ F L loc ( R ) is an almost periodic function witha summable Fourier series if h ( ξ ) = X n ∈ Z c n e iξt n , X n ∈ Z | c n | < ∞ , ξ, t n ∈ R , n ∈ Z . (3.2)The set of all such functions is denoted by AP or AP ( R ).We note that AP is a Banach space with the norm k h k AP = X n ∈ Z | c n | , where h ∈ AP is given by (3.2). We also mention [9, Proposition 2.11], whichstates that for such h we haveˇ T ( h ) = X n ∈ Z c n T ( t n ) ∈ B ( X ) . (3.3) Proposition 3.10.
Any function h ∈ AP is X -regular for any X . Moreover,if λ / ∈ h ( R ) , then (cid:13)(cid:13)(cid:13)(cid:0) λ − ˇ T ( h ) (cid:1) − (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) λ − h (cid:13)(cid:13)(cid:13)(cid:13) AP . Proof.
Let R d be the group of real numbers with the discrete topology and R c be its Pontryagin dual – the Bohr compactification of R . It is well-knownthat a function in AP ( R ) has a unique continuous extension to R c andcan be identified with an element of F L ( R d ) – the Fourier algebra of thelosed Operator Functional Calculus 9group R d . The closure of Λ( X , T ) in R c is then a compact subset of R c and, given λ / ∈ h (Λ( X , T )), the Wiener-L´evy theorem for locally compactAbelian groups ([25, Theorem 6.1.1]) establishes existence of g λ ∈ AP ( R )that satisfies (3.1) in a neighborhood of Λ( X , T ) in R . Hence, X -regularityfollows from (3.3), i.e. [9, Proposition 2.11].For λ / ∈ h ( R ), it suffices to apply the almost periodic version of Wiener’s1 /f lemma [2, 23] that shows that g λ = λ − h ∈ AP . The desired estimatethen follows from (3.3). Example . Let h ( ξ ) = e iξt for some t ∈ R and λ ∈ C \ T , where T = { z ∈ C : | z | = 1 } . From (3.3), we get ˇ T ( h ) = T ( t ). Using the estimate fromProposition 3.10, we get by direct computation that (cid:13)(cid:13)(cid:13) ( λ − T ( t )) − (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13) λ − h (cid:13)(cid:13)(cid:13)(cid:13) AP = 1 | − | λ || = (dist( λ, T )) − . In general, it may be hard to check if a given function is X -regular. Wecite [7] and references therein for related results. We note that the notion ofregularity at infinity discussed in [7] is more restrictive than X -regularity.The following theorem provides a special case when the assumption of X -regularity is not needed. Theorem 3.11.
Assume that X = H is a Hilbert space and the representation T is unitary. Then for any h ∈ F L loc ( R ) we have σ ( ˇ T ( h )) = h (Λ( X , T )) .Moreover, given λ / ∈ h (Λ( X , T )) , we have (cid:13)(cid:13) ( λ − ˇ T ( h )) − (cid:13)(cid:13) = (cid:16) dist (cid:16) λ, h (Λ( X , T ) (cid:17)(cid:17) − . (3.4) Proof.
In view of Lemma 3.3, we only need to prove σ ( ˇ T ( h )) ⊆ h (Λ( X , T )), h ∈ F L loc ( R ). Pick λ / ∈ h (Λ( X , T )). We will show that λ ∈ ρ ( ˇ T ( h )).Let ( φ a ) be a cf-b.a.i. and X a = T ( φ a ) X , a >
0, be the correspondingsubmodules of X . From (2.4), we have Λ( X a , T ) ⊆ supp b φ a ∩ Λ( X , T ). Hence, X a ⊆ X comp ⊆ D ( ˇ T ( h )) and (2.3) implies that X a is invariant for ˇ T ( h ).Therefore, the restrictions of ˇ T ( h ) to X a , a >
0, are well defined. We willdenote these restrictions by B a .Since h is X a -regular, Theorem 3.5 applies for B a yielding σ ( B a ) = h (Λ( X a , T )) ⊆ h (Λ( X , T )). It follows that λ ∈ ρ ( B a ). Moreover, since therepresentation T is unitary, the operators B a are normal. Therefore, thenorms of their resolvents satisfy k R ( λ ; B a ) k = (dist( λ, σ ( B a )) − ≤ (cid:16) dist (cid:16) λ, h (Λ( X , T ) (cid:17)(cid:17) − . (3.5)Now, since the representation T is strongly continuous, Proposition 2.2 im-plies that x = lim a →∞ T ( φ a ) x x ∈ X . From (3.5) and the Banach-Steinhaus theorem, weget that C = C λ given by Cx = lim a →∞ R ( λ ; B a ) T ( φ a ) x (3.6)is a well-defined bounded linear operator. By direct computation, it followsthat T ( f ) C ( λ − ˇ T ( h )) x = T ( f )( λ − ˇ T ( h )) Cx, for any x ∈ D ( ˇ T ( h )) and f ∈ L ( R ) with supp b f compact. Since the module X is non-degenerate, we get C = ( λ − ˇ T ( h )) − . Finally, the equality in (3.4)follows since C = ( λ − ˇ T ( h )) − is a normal operator. Remark . Often [1, 4, 19] a representation T and operators of the formˇ T ( h ) act not just in a single Banach module but in a whole chain ( X p ) ofsuch modules. For example, matrices with sufficient off-diagonal decay definebounded operators on all ℓ p , p ∈ [1 , ∞ ). In this case, it is not unusual for σ ( ˇ T ( h )) to be independent of p . If also one of the modules X p happened to bea Hilbert space, Theorem 3.11 would then yield a spectral mapping theoremfor all Banach modules X p in the chain.The proof of Theorem 3.11 leads us to define the following notion ofregularity for functions in F L loc ( R ). Definition 3.3.
Let ( ϕ a ) be the cf-b.a.i. from Example 2.1. A function h ∈F L loc ( R ) is called spectrally admissible if for each λ / ∈ h (Λ( X , T )) there existfunctions g aλ ∈ F L ( R ) such that g aλ = b ϕ a λ − h in a neighborhood of h (Λ( X , T ))and M h ( λ ) := sup a> (cid:13)(cid:13) F − ( g aλ ) (cid:13)(cid:13) < ∞ . (3.7) Remark . In the above definition, instead of the functions from Example2.1 we may use a cf-b.a.i. ( ϕ a,n ), n >
1, given by b ϕ a,n ( ξ ) = τ a,n ( ξ ) = , | ξ | ≤ a, n − a ( na − | ξ | ) , a < | ξ | < na, , | ξ | ≥ na. From [25, Proposition 5.1.5] we get k ϕ a,n k ≤ q n +1 n − , which may give asmaller M h ( λ ) in (3.7). We also note that Lemma 2.1 may often be used toprove spectral admissibility. Theorem 3.12.
Let ( X , T ) be a non-degenerate Banach L ( R ) -module suchthat the representation T is strongly continuous. Assume that a function h ∈F L loc ( R ) is spectrally admissible. Then σ ( ˇ T ( h )) = h (Λ( X , T )) . Moreover,given λ / ∈ h (Λ( X , T )) , we have (cid:13)(cid:13) ( λ − ˇ T ( h )) − (cid:13)(cid:13) ≤ M h ( λ ) . losed Operator Functional Calculus 11 Proof.
As in the proof of Theorem 3.11, given a cf-b.a.i. ( φ a ), we let X a = T ( φ a ) X and B a = ˇ T ( h ) | X a . Observe that for a sufficiently large b > g bλ from Definition 3.3 satisfies g bλ ( ξ ) = λ − h ( ξ ) for every ξ in aneighborhood of h (Λ( X a , T )). From Theorem 3.5, we deduce that k R ( λ, B a ) k ≤ (cid:13)(cid:13) F − (cid:0) g bλ (cid:1)(cid:13)(cid:13) ≤ M h ( λ ) < ∞ , λ / ∈ h (Λ( X , T )) . The remainder of the proof of Theorem 3.11 now goes through in this setting.An application of the Banach-Steinhaus theorem shows that (3.6) definesan operator C λ ∈ B ( H ) satisfying k C λ k ≤ M h ( λ ) sup a k φ a k , and, choosing φ a = γ a defined by (2.9) gives k C λ k ≤ M h ( λ ). It is then verified by directcomputation that C λ = ( λ − ˇ T ( h )) − .
4. Spectral estimates for the operator L . In this section, we prove Theorem 1.1. The approach we pursue is based onthe following result which holds for Hilbert-Schmidt perturbations of generalself-adjoint operators on an abstract complex Hilbert space H . The ideal ofall Hilbert-Schmidt operators in H will be denoted by S ( H ). Theorem 4.1.
Let A : D ( A ) ⊆ H → H be a self-adjoint operator and B ∈ S ( H ) . Then there exists a continuous real-valued function f ∈ L ( R ) suchthat for any λ ∈ σ ( A + B ) one has |ℑ mλ | ≤ f ( ℜ eλ ) . We believe that the above result has been known for a long time. Sincewe didn’t find the reference, however, we provide its proof in Section 5. Wecite [18] for related results.Clearly, the perturbation V of the form (1.1) may not be Hilbert-Schmidt; it is not even a bounded operator, in general. We will, however,construct a similarity transform which will allow us to use the above result. Definition 4.1.
Two linear operators A i : D ( A i ) ⊂ H → H , i = 1 ,
2, arecalled similar if there exists an invertible operator U ∈ B ( H ) such that U D ( A ) = D ( A ) and A U x = U A x , x ∈ D ( A ). We call the operator U the similarity transform of A into A .It is immediate that for similar operators A and A one has σ ( A ) = σ ( A ). Thus, to prove Theorem 1.1, it suffices to construct a similarity trans-form of L into − i ddt − B with B ∈ S ( H ). In order to do it, we apply the F L loc functional calculus in the space X = L A ( H ) of closed linear A -boundedoperators that is defined as follows. Definition 4.2.
Let A : D ( A ) ⊂ H → H be a closed linear operator. Alinear operator X : D ( X ) ⊂ H → H is A -bounded if D ( X ) ⊇ D ( A ) and k X k A = inf { c > k Xx k ≤ c ( k x k + k Ax k ) , x ∈ D ( A ) } < ∞ .The space L A ( H ) of all A -bounded linear operators with the domainequal to D ( A ) is a Banach space with respect to the norm k · k A . For denselydefined operators A , restricting the domain of bounded operators to D ( A ),allows us to view B ( H ) as a subspace of L A ( H ).2 Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. UskovaNow we need to define the Banach module structure in X = L A ( H )with A = − i ddt . We begin with a Banach module structure in H = L ( R ).The operator A = − i ddt : W , ( R ) ⊆ L ( R ) → L ( R ) is self-adjoint,and the operator iA generates an isometric strongly continuous group oftranslations T : R → B ( H ), T ( t ) x ( s ) = x ( t + s ), x ∈ L ( R ). The non-degenerate L -module structure in H is then given by convolution: T ( f ) x = Z R f ( t ) T ( − t ) xdt, x ∈ H. Next, we let T : R → B ( X ) = B ( L A ( H )) be defined by T ( t ) X = T ( t ) XT ( − t ), t ∈ R . Since T is an isometric representation, we get that T also has thisproperty. We then have that( T ( f ) X ) x = Z R f ( t )( T ( − t ) X ) xdt = Z R f ( t ) T ( − t ) XT ( t ) xdt, (4.1) X ∈ X , x ∈ D ( A ) , f ∈ L ( R ) , defines a non-degenerate L -module structurein X that is associated with the representation T . Moreover, the generator A of the module ( X , T ) satisfies A X = AX − XA, X ∈ D ( A ) , see e.g. [16]. We now apply (2.8) in this Banach module ( X , T ) to get A ( T ( ψ a ) X ) − ( T ( ψ a ) X ) A = X − T ( ϕ a ) X, X ∈ D ( A ) , (4.2)where the functions ϕ a and ψ a , a >
0, are defined in Examples 2.1 and 2.2.Moreover, the discussion following (2.8) shows that for any X ∈ L A ( H ) and x ∈ D ( A ) we have A ( T ( ψ a ) X ) x − ( T ( ψ a ) X ) Ax = Xx − ( T ( ϕ a ) X ) x. (4.3) Lemma 4.2.
Consider the functions ϕ a and ψ a , a > , defined in Examples2.1 and 2.2. An operator V of the form (1.1) has the following properties.1. T ( ϕ a ) V ∈ S ( H ) and kT ( ϕ a ) V k = q a π k v k .2. T ( ψ a ) V ∈ S ( H ) and kT ( ψ a ) V k = q − ln 2 aπ k v k .3. T ( ψ a ) V ( W , ( R )) ⊆ W , ( R ) .4. V T ( ψ a ) V ∈ S ( H ) and k V T ( ψ a ) V k ≤ π +1 π √ k v k .5. Given ǫ > , there is λ ǫ ∈ C \ R such that k V ( λ ǫ − A ) − k < ǫ .Proof. Observe that for any h ∈ L ∩ L we have( T ( h ) V ) x ( s ) = Z R h ( t ) v ( s − t ) x ( − s + 2 t ) dt. (4.4)Hence, T ( h ) V ∈ S ( H ) and kT ( h ) V k = 12 Z R Z R | h ( t ) v ( s − t ) | dsdt = 12 k h k k v k . Plugging in the norms k ϕ a k and k ψ a k from Examples 2.1 and 2.2 establishesProperties 1 and 2.losed Operator Functional Calculus 13To prove Property 3, pick z > R = R ( z ; A ) = ( z − A ) − . Usingthe definition of the generator of a Banach module, we have Rx = Z R f z ( t ) T ( − t ) xdt, x ∈ H, (4.5)where b f z ( λ ) = ( λ − z ) − . Then for any h ∈ L ∩ L , letting h t = T ( t ) h , oneeasily gets ( T ( h ) V ) Rx = R ( T ( h t ) V ) x, after plugging in (4.1) and (4.5). Hence, Property 3 follows.Next, observe that for any h ∈ L ∩ L we have V ( T ( h ) V ) x ( s ) = Z R v ( s ) h ( t ) v ( − s − t ) x ( s + 2 t ) dt. (4.6)Hence, V T ( h ) V ∈ S ( H ) and k V T ( h ) V k = 12 Z R Z R | v ( s ) h ( t ) v ( − s − t ) | dsdt ≤ k h k ∞ k v k , and the estimate for k ψ a k ∞ from Example 2.2 yields Property 4.Finally, observe that (4.5) yields V R ( λ ǫ ; A ) x ( s ) = Z R v ( s ) f iλ ǫ ( t ) x ( − s − t ) dt, x ∈ H, λ ǫ ∈ R \ { } , and, hence, k V R ( λ ǫ ; A ) k = 12 π k v k Z R dt | t − iλ ǫ | = 12 λ ǫ k v k , which implies Property 5.From the above lemma, it is clear that we can choose a > kT ( ψ a ) V k <
1. Then operator U = I + T ( ψ a ) V ∈ B ( H ) is invertible andthe estimates in the lemma together with (4.3) allow us to use [12, Theorem3.3] to obtain the following result. Theorem 4.3.
Consider an operator L with V of the form (1.1) and thefunctions ϕ a and ψ a , a > , defined in Examples 2.1 and 2.2. Pick a =4 − ln 2 π k v k . Then kT ( ψ a ) V k = , U = I + T ( ψ a ) V ∈ B ( H ) , U − ∈ B ( H ) ,and k U − − I k ≤ . Moreover, U is the similarity transform of L into − i ddt − B , where B = T ( ϕ a ) V + U − ( V T ( ψ a ) V − ( T ( ψ a ) V ) T ( ϕ a ) V ) , = U − ( V T ( ψ a ) V + T ( ϕ a ) V ) ∈ S ( H ) , and we have k B k ≤ √ π (cid:18) q − ln 23 + π + 1 (cid:19) k v k ≤ . k v k .Proof. Even though the assumptions of [12, Theorem 3.3] are slightly differ-ent, its proof applies nearly verbatim to establish the similarity of L and − i ddt − B . The postulated estimates are then easily obtained by direct com-putation.4 Anatoly G. Baskakov, Ilya A. Krishtal and Natalia B. UskovaTheorems 4.1 and 4.3 immediately yield Theorem 1.1. Remark . We observe that analogs of Theorem 1.1 hold for any self-adjointoperator A and a perturbation V ∈ L A ( H ) for which the properties of Lemma4.2 hold without the specific estimates of the Hilbert-Schmidt norms. More-over, Properties 1, 2, and 4 may be replaced by the following weaker assump-tions: • T ( ψ a ) V ∈ B ( H ) and there is a > kT ( ψ a ) V k < • V T ( ψ a ) V + T ( ϕ a ) V ∈ S ( H ).
5. Appendix.
In this section, we collect the proofs that we include for completeness of theexposition.
Proof of Theorem 4.1.
Let E n = E ([ − n, n ]) be the spectral projection corre-sponding to A and the interval [ − n, n ], n ∈ N , and e E n = I − E n . Similarly, let A n = E n A = AE n , e A = A − A n , B n = E n BE n and e B n = B − B n . Observethat k B k = k B n k + k e B n k , and the sequence ( b n ) with b n = k e B n k , n ∈ N ,is in ℓ ( N ). Observe also that for λ ∈ C \ R such that |ℜ eλ | ≥ n + 2 k B k or |ℑ mλ | ≥ k B k we have( λ − A − B n ) − = ( λ − A ) − e E n + E n ( λ − A − B n ) − E n = ( λ − A ) − e E n + ∞ X k =0 (cid:0) B n ( λ − A n ) − (cid:1) k ! , where the series converges absolutely since (cid:13)(cid:13)(cid:0) B n ( λ − A n ) − (cid:1)(cid:13)(cid:13) ≤ due todist( λ, σ ( A n )) > k B k ≥ k B n k . Hence for λ ∈ C \ R with dist( λ, [ − n, n ]) > k B k we have (cid:13)(cid:13) ( λ − A − B n ) − (cid:13)(cid:13) ≤ |ℑ mλ | ∞ X k =0 − k ! = 3 |ℑ mλ | . For any n ∈ N let Q n = (cid:8) λ ∈ C : |ℑ mλ | > k e B n k and |ℑ mλ | > k B k , if |ℜ eλ | ≤ n + 2 k B k (cid:9) . Then for any λ ∈ Q n we have( λ − A − B ) − = ( λ − A − B n ) − (cid:16) I − e B n ( λ − A − B n ) − (cid:17) − = ( λ − A − B n ) − ∞ X k =0 (cid:16) e B n ( λ − A − B n ) − (cid:17) k ! ∈ B ( H ) , and the result follows by considering the union of Q n , n ∈ N .losed Operator Functional Calculus 15 Remark . We note that for an explicitly known operator B the aboveproof essentially yields an algorithm for constructing a function f ∈ L thatenvelops the spectrum σ ( A + B ). Proof of Lemma 2.1.
Observe that for any a > k f k = Z (cid:12)(cid:12)(cid:12)(cid:12) a + it ( a + it ) f ( t ) (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ (cid:18)Z dta + t (cid:19) · (cid:18)Z | ( a + it ) f ( t ) | dt (cid:19) ≤ r πa (cid:18) a k f k + 1 √ π k b f ′ k (cid:19) = 1 √ (cid:18) √ a k b f k + 1 √ a k b f ′ k (cid:19) , by the Cauchy-Schwarz inequality. Plugging in a = k b f ′ k k b f k , yields the desiredresult. Acknowledgement
The first and third authors were supported in part by the RFBR grant 19-01-00732.
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Ilya A. KrishtalDepartment of Mathematical SciencesNorthern Illinois UniversityDeKalb, IL 60115USAe-mail: [email protected]