Resolvent estimates for operators belonging to exponential classes
aa r X i v : . [ m a t h . F A ] S e p RESOLVENT ESTIMATES FOR OPERATORS BELONGING TOEXPONENTIAL CLASSES
OSCAR F. BANDTLOW
Abstract.
For a, α > E ( a, α ) be the set of all compact operators A on a separable Hilbert space such that s n ( A ) = O (exp( − an α )), where s n ( A )denotes the n -th singular number of A . We provide upper bounds for thenorm of the resolvent ( zI − A ) − of A in terms of a quantity describing thedeparture from normality of A and the distance of z to the spectrum of A .As a consequence we obtain upper bounds for the Hausdorff distance of thespectra of two operators in E ( a, α ). Introduction
Let A and B be compact operators on a Hilbert space. It is known that if k A − B k is small then the spectra of A and B are close in a suitable sense (forexample, with respect to the Hausdorff metric on the space of compact subsets of C ). Just how close are they? Standard perturbation theory gives bounds in termsof quantities that require a rather detailed knowledge of the spectral properties ofboth operators, for example the norms of the resolvents of A and B along contoursin the complex plane, which are difficult to obtain in practice.The main concern of this article is to derive an upper bound for the norm of theresolvent ( zI − A ) − of an operator A belonging to certain subclasses of compactoperators in terms of simple, readily computable quantities, typically involving thedistance of z to the spectrum of A and a number measuring the departure fromnormality of A . As a result, we obtain simple upper bounds for the Hausdorffdistance of the spectra of two operators in these subclasses. Estimates of this typehave previously been obtained for operators in the Schatten classes (see [Gil, Ban])and more generally (but less sharp), for operators belonging to Φ-ideals (see [Pok]).These subclasses, termed exponential classes , are constructed as follows. For a, α > E ( a, α ) denote the collection of all compact operators on a separableHilbert space for which s n ( A ) = O (exp( − an α )), where s n ( A ) denotes the n -th sin-gular number of A . As we shall see, E ( a, α ) is not a linear space (see Remark 2.9),hence a fortiori not an operator ideal, and may thus be viewed as a slightly patho-logical object in this context. There is nevertheless compelling reason to considerthese classes: on the one hand, the resolvent bounds given in [Ban, Gil, Pok], whileapplicable to operators in E ( a, α ), can be improved significantly (see Remark 3.2),the lack of linear structure posing almost no problem for the derivation of theseimprovements. On the other hand, operators belonging to exponential classes arise Date : 14 July 2008.1991
Mathematics Subject Classification.
Primary 47A10; Secondary 47B06, 47B07.
Key words and phrases.
Resolvent growth, exponential class, departure from normality, boundsfor spectral distance. naturally in a number of different ways. For example, if A is an integral operatorwith real analytic kernel given as a function on [0 , d × [0 , d , then A ∈ E ( a, /d )for some a > E ( a, /d ) for some a > C d whose symbols are strict contractions, or more generally transferoperators corresponding to holomorphic map-weight systems on C d , the latter pro-viding one of the motivations to look more closely into the properties of operatorsbelonging to exponential classes (see Example 2.3 (iv) and [BanJ1, BanJ2, BanJ3]).This article is organised as follows. In Section 2 we define the exponential classesand study some of their properties. In particular, we shall give a sharp descriptionof the behaviour of exponential classes under addition (see Proposition 2.8), anda sharp characterisation of the eigenvalue asymptotics of an operator in a givenexponential class (see Proposition 2.10). Some of the arguments in this section relyon results concerning monotonic arrangements of sequences, which are presented inthe Appendix. In Section 3 we will use techniques similar to those already employedin [Ban] to obtain resolvent estimates for operators in E ( a, α ) (see Theorem 3.13).In particular we shall give a sharp estimate for the growth of the resolvent of aquasi-nilpotent operator in E ( a, α ) (see Proposition 3.1). These estimates will thenbe used in the final section to deduce Theorem 4.2, which provides spectral variationand spectral distance formulae for operators in E ( a, α ). Notation 1.1.
Throughout this article H and H i will be assumed to be separableHilbert spaces. We use L ( H , H ) to denote the Banach space of bounded linear op-erators from H to H equipped with the usual norm and S ∞ ( H , H ) ⊂ L ( H , H )to denote the closed subspace of compact operators from H to H . We shall oftenwrite L or S ∞ if the Hilbert spaces H and H are understood.For A ∈ S ∞ ( H , H ) we use s k ( A ) := inf { k A − F k : F ∈ L ( H , H ) , rank( F ) < k } ( k ∈ N )to denote the k -th approximation number of A , and s ( A ) to denote the sequence { s k ( A ) } ∞ k =1 .The spectrum and the resolvent set of A ∈ L ( H, H ) will be denoted by σ ( A )and ̺ ( A ), respectively. For A ∈ S ∞ ( H, H ) we let λ ( A ) = { λ k ( A ) } ∞ k =1 denote thesequence of eigenvalues of A , each eigenvalue repeated according to its algebraicmultiplicity, and ordered by magnitude, so that | λ ( A ) | ≥ | λ ( A ) | ≥ . . . . Similarly,we write | λ ( A ) | for the sequence {| λ k ( A ) |} ∞ k =1 .We note that the approximation numbers coincide with the singular numbers ,that is, s k ( A ) = p λ k ( A ∗ A ) ( k ∈ N ) , where A ∗ ∈ L ( H , H ) denotes the adjoint of A ∈ L ( H , H ). For more informationabout these notions see, for example, [Pie, GK, DS, Rin].2. Exponential classes
Exponential classes arise by grouping together all operators A whose singularnumbers s n ( A ) decay at a given (stretched) exponential rate, that is, s n ( A ) = O (exp( − an α )) for fixed a > α >
0. Our main concern in this section willbe to investigate how these classes behave under addition and multiplication, andto determine the rate of decay of the eigenvalue sequence of an operator in a given
ESOLVENT ESTIMATES 3 class. Some of the arguments in this section depend on results concerning monotonicarrangements of sequences, which are discussed in the Appendix.
Definition 2.1.
Let a > α >
0. Then E ( a, α ) := (cid:26) x ∈ C N : | x | a,α := sup n ∈ N | x n | exp( an α ) < ∞ (cid:27) , and E ( a, α ; H , H ) := (cid:26) A ∈ S ∞ ( H , H ) : | A | a,α := sup n ∈ N s n ( A ) exp( an α ) < ∞ (cid:27) . are called exponential classes of type ( a, α ) of sequences and operators , respectively.The numbers | x | a,α and | A | a,α are called ( a, α ) -gauge or simply gauge of x and A ,respectively.Whenever the Hilbert spaces are clear from the context, we suppress referenceto them and simply write E ( a, α ) instead of E ( a, α ; H , H ). Remark 2.2.
Note that E ( a, α ) is a Banach space when equipped with the gauge |·| a,α . On the other hand, the set E ( a, α ), the non-commutative analogue of E ( a, α ),is not even a linear space in general (see Proposition 2.8 and Remark 2.9 below).The reason for this is that if a sequence lies in E ( a, α ) then a rearrangement ofthis sequence need not; in particular E ( a, α ) is not a Calkin space in the sense of[Sim2, p. 26] (cf. also [Cal]). However, E ( a, α ; H, H ) turns out to be a pre-ideal (see Remark 2.6).Operators belonging to exponential classes arise naturally in a number of differ-ent contexts.
Example 2.3. (i) Let σ be a complex measure on the circle group T such that its Fouriertransform satisfies | b σ ( n ) | ≤ exp( − a | n | ) ( n ∈ Z ) . It is not difficult to see that this is the case if and only if σ is absolutely contin-uous with respect to Haar measure on T and the corresponding Radon-Nikod´ymderivative is holomorphic on T .Let L ( T ) be the complex Hilbert space of square-integrable functions on T ,with respect to Haar measure on T . Let A : L ( T ) −→ L ( T ) be the convolutionoperator Af = f ∗ σ . The spectrum of A is b σ ( Z ) ∪ { } and the spectrum of A ∗ A equals [ σ ∗ e σ ( Z ) ∪ { } = | b σ ( Z ) | ∪ { } , where d e σ ( t ) = dσ ( t − ) (cf. [BerF, p. 87]). Moreover, A is a compactoperator and the non-zero eigenvalues of A are precisely the numbers b σ ( n ) for n ∈ Z . In order to locate A in the scale of exponential classes, we enumerate theeigenvalues of A as follows x n = b σ (cid:18) ( − n n + ( − n − (cid:19) ( n ∈ N ) . O.F. BANDTLOW
Then the sequence x belongs to the class E ( a/ ,
1) with | x | / , ≤ exp( a/
2) since (cid:12)(cid:12)(cid:12)(cid:12)b σ (cid:18) ( − n n + ( − n − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ exp (cid:18) − a (cid:12)(cid:12)(cid:12)(cid:12) ( − n n + ( − n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ≤ exp (cid:16) − a n − (cid:17) = exp (cid:16) a (cid:17) exp (cid:16) − a n (cid:17) . By Corollary 5.4 the decreasing arrangement x (+) of x also belongs to E ( a/ , | x (+) | / , ≤ exp( a/ s ( A ) = | λ ( A ) | ∈ E ( a/ ,
1) and it follows that A ∈ E ( a/ ,
1) with | A | a/ , ≤ exp( a/ A is an integral operator on the space of Lebesgue square-integrablefunctions on the d -dimensional unit-cube [0 , d whose kernel is real analytic on[0 , d × [0 , d , then A ∈ E (1 /d ).(iii) For a domain Ω ⊂ R d with d > h (Ω) be the Bergman space of Lebesguesquare-integrable harmonic functions on Ω. If Ω , Ω ⊂ R d are two domains suchthat Ω is compactly contained in Ω , that is Ω is a compact subset of Ω , thenthe canonical embedding J : h (Ω ) ֒ → h (Ω ) given by Jf = f | Ω belongs tothe exponential class E (1 / ( d − and Ω with simplegeometries it is possible to sharply locate J in an exponential class E ( a, / ( d − a, / ( d − J exactly. See [BanC].(iv) For Ω ⊂ C d a bounded domain, let L Hol (Ω) denote the Bergman space ofholomorphic functions which are square-integrable with respect to 2 d -dimensionalLebesgue measure on Ω. Given a collection { φ , . . . , φ K } of holomorphic maps φ k : Ω → Ω and a collection { w , . . . , w K } of bounded holomorphic functions w k : Ω → C consider the corresponding linear operator A on L Hol (Ω) given by Af := K X k =1 w k · f ◦ φ k . If ∪ k φ k (Ω) is compactly contained in Ω (see the previous example for the definition)then A is a compact endomorphism of L Hol (Ω) and A ∈ E ( a, /d ), where a dependson the geometry of Ω and ∪ k φ k (Ω) (see [BanJ3]).Operators of this type, known as transfer operators , play an important role inthe ergodic theory of expanding dynamical systems due to the remarkable factthat their spectral data can be used to gain insight into geometric and dynamicinvariants of a given expanding dynamical system (see [Rue]). As a consequence, itis of interest to determine spectral properties of these operators exactly, or at leastto a given accuracy. The latter problem, namely that of calculating rigorous errorbounds for spectral approximation procedures for these operators provided one ofthe main motivations to study operators in exponential classes (see [BanJ3]).We shall now study some of the properties of the classes E ( a, α ). First we notethat if we order the indices ( a, α ) reverse lexicographically, that is, by defining( a, α ) ≺ ( a ′ , α ′ ) : ⇔ ( α < α ′ ) or ( α = α ′ and a < a ′ ) , then we obtain the following inclusions. Proposition 2.4.
Let a, a ′ > and α, α ′ > . Then ESOLVENT ESTIMATES 5 (i) ( a, α ) ≺ ( a ′ , α ′ ) ⇔ E ( a ′ , α ′ ) $ E ( a, α ) ; (ii) ( a, α ) ≺ ( a ′ , α ′ ) ⇔ E ( a ′ , α ′ ) $ E ( a, α ) .Proof. The proof of (i) is straightforward and will be omitted. Assertion (ii) followsfrom (i) together with the observation that A ∈ E ( a, α ) iff s ( A ) ∈ E ( a, α ) and thefact that for every monotonically decreasing x ∈ E ( a, α ) there is a compact A with s ( A ) = x . (cid:3) While E ( a, α ) is not, in general, a linear space, it does enjoy the following closureproperties. Proposition 2.5.
Let a, α > . If A ∈ L ( H , H ) , B ∈ E ( a, α ; H , H ) , and C ∈ L ( H , H ) , then | ABC | a,α ≤ k A k | B | a,α k C k . In particular, L ( H , H ) E ( a, α ; H , H ) L ( H , H ) ⊂ E ( a, α ; H , H ) . Proof.
Follows from s k ( ABC ) ≤ k A k s k ( B ) k C k for k ∈ N (see [Pie, 2.2]). (cid:3) Remark 2.6.
The proposition implies that L ( H, H ) E ( a, α ; H, H ) L ( H, H ) ⊂ E ( a, α ; H, H ) . Thus the classes E ( a, α ; H, H ), while lacking linear structure, satisfy part of thedefinition of an operator ideal. In other words, E ( a, α ; H, H ) is what is sometimesreferred to as a pre-ideal (see, for example, [Nel]).We now consider in more detail the relation between different exponential classesunder addition. We start with a general result concerning the singular numbers ofa sum of operators.
Proposition 2.7.
Let A k ∈ S ∞ ( H , H ) for ≤ k ≤ K . Then s n K X k =1 A k ! ≤ Kσ n ( n ∈ N ) , where σ denotes the decreasing arrangement (see the Appendix) of the K singularnumber sequences s ( A ) , . . . , s ( A K ) .Proof. Set A := P Kk =1 A k . The compactness of the A k means they have Schmidtrepresentations A k = ∞ X l =1 s l ( A k ) a ( k ) l ⊗ b ( k ) l , where { a ( k ) l } l ∈ N and { b ( k ) l } l ∈ N are suitable orthonormal systems in H and H respectively. Here a ⊗ b , where a ∈ H and b ∈ H , denotes the rank-1 operator H → H given by ( a ⊗ b ) x = ( x, a ) H b . Let ν : N → N and κ : N → N be functions that effect the decreasing arrangementof the singular numbers of the A k in the sense that σ n = s ν ( n ) ( A κ ( n ) ) ( n ∈ N ) . O.F. BANDTLOW
Then A = ∞ X n =1 σ n a ( κ ( n )) ν ( n ) ⊗ b ( κ ( n )) ν ( n ) , which suggests defining, for each m ∈ N , the rank- m operator F m : H → H by F := 0 ,F m := m X n =1 σ n a ( κ ( n )) ν ( n ) ⊗ b ( κ ( n )) ν ( n ) ( m ∈ N ) . If x ∈ H and y ∈ H then | (( A − F m − ) x, y ) H | ≤ ∞ X n = m σ n (cid:12)(cid:12)(cid:12) ( x, a ( κ ( n )) ν ( n ) ) H ( b ( κ ( n )) ν ( n ) , y ) H (cid:12)(cid:12)(cid:12) ≤ σ m ∞ X n =1 (cid:12)(cid:12)(cid:12) ( x, a ( κ ( n )) ν ( n ) ) H ( b ( κ ( n )) ν ( n ) , y ) H (cid:12)(cid:12)(cid:12) = σ m K X k =1 ∞ X l =1 (cid:12)(cid:12)(cid:12) ( x, a ( k ) l ) H ( b ( k ) l , y ) H (cid:12)(cid:12)(cid:12) ≤ σ m K X k =1 vuut ∞ X l =1 (cid:12)(cid:12)(cid:12) ( x, a ( k ) l ) H (cid:12)(cid:12)(cid:12) vuut ∞ X l =1 (cid:12)(cid:12)(cid:12) ( b ( k ) l , y ) H (cid:12)(cid:12)(cid:12) ≤ σ m K X k =1 k x k H k y k H = σ m K k x k H k y k H . This estimate justifies the rearrangements (since the series are absolutely conver-gent) and also yields k A − F m − k ≤ Kσ m , from which the assertion follows. (cid:3) Proposition 2.8.
Suppose that A n ∈ E ( a n , α ; H , H ) for ≤ n ≤ K . Let A := P Kn =1 A n and a ′ := ( P Kn =1 a − /αn ) − α . Then (i) A ∈ E ( a ′ , α ) with | A | a ′ ,α ≤ K max ≤ n ≤ K | A n | a n ,α . In particular E ( a , α ) + · · · + E ( a K , α ) ⊂ E ( a ′ , α ) . (ii) If both H and H are infinite-dimensional then the inclusion above is sharpin the sense that E ( a , α ) + · · · + E ( a K , α ) E ( b, α ) , whenever b > a ′ .Proof. Assertion (i) follows from Proposition 2.7 and Corollary 5.4 (i), which givesan upper bound on the rate of decay of the decreasing arrangement of the sequences s ( A ) , . . . , s ( A K ).For the proof of (ii) define K sequences s (1) , . . . , s ( K ) by s ( k ) n := exp( − a k n α ) ( n ∈ N ) . It turns out that it suffices to exhibit K compact operators A k with s n ( A k ) = s ( k ) n such that s ( A ) is the decreasing arrangement of the sequences s (1) , . . . , s ( K ) . Tosee this, note that then A k ∈ E ( a k , α ) for k ∈ { , · · · , K } . At the same time ESOLVENT ESTIMATES 7 s n ( A ) ≥ exp( − a ′ ( n + K ) α ) by Corollary 5.4 (ii), so that A E ( b, α ) whenever b > a ′ .In order to construct these operators we proceed as follows. Since each H i wasassumed to be infinite-dimensional we can choose an orthonormal basis { h ( i ) n } n ∈ N for each of them. For each k = 1 , . . . , K , we now define a compact operator A k : H → H by A k h (1) n := ( s ( k )( n +( k − /K h (2) n for n ∈ K N − ( k − , n K N − ( k − . It is not difficult to see that s n ( A k ) = s ( k ) n . Moreover, it is easily verified thatthe singular numbers of A are precisely the numbers of the form s ( k ) n with n ∈ N and k = 1 , . . . , K . Thus, s ( A ) is the decreasing arrangement of the sequences s (1) , . . . , s ( K ) as required. (cid:3) Remark 2.9.
The proposition implies that E ( a, α ) + E ( a, α ) ⊂ E (2 − α a, α ), but E ( a, α ) + E ( a, α ) E ( a, α ), because 2 − α a < a . In particular, E ( a, α ) is not alinear space.The following result establishes a sharp bound on the eigenvalue decay rate ineach exponential class. Proposition 2.10.
Let a, α > and A ∈ E ( a, α ; H, H ) . Then λ ( A ) ∈ E ( a/ (1 + α ) , α ) with | λ ( A ) | a/ (1+ α ) ,α ≤ | A | a,α . If H is infinite-dimensional, the result is sharp in the sense that there is an operator A ∈ E ( a, α ; H, H ) such that λ ( A )
6∈ E ( b, α ) whenever b > a/ (1 + α ) .Proof. If A ∈ E ( a, α ) then s k ( A ) ≤ | A | a,α exp( − ak α ). Using the multiplicativeWeyl inequality [Pie, 3.5.1] we have(1) | λ k ( A ) | k ≤ k Y l =1 | λ l ( A ) | ≤ k Y l =1 s l ( A ) ≤ k Y l =1 | A | a,α exp( − al α ) == | A | ka,α exp( − a k X l =1 l α ) . But P kl =1 l α ≥ R k x α dx = α k α +1 , which combined with (1) yields | λ k ( A ) | ≤ | A | a,α exp( − ak α / (1 + α )) . Sharpness is proved in several steps. We start with the following observation. Let τ ≥ . . . ≥ τ N ≥ C ( τ , . . . , τ N ) ∈ L ( C N , C N ) given by C ( τ , . . . , τ N ) := τ . . . . . . . . . τ N − τ N . . . . O.F. BANDTLOW
It easy to see that s n ( C ( τ , . . . , τ N )) = τ n and | λ ( C ( τ , . . . , τ N )) | = · · · = | λ N ( C ( τ , . . . , τ N )) | = ( τ · · · τ N ) /N . The desired operator is constructed as follows. Fix a > α >
0. Nextchoose a super-exponentially increasing sequence N n , that is, N n is increasing andlim n →∞ N n − /N n = 0. For definiteness we could set N n = exp( n ).Put N = 0 and define d n := N n − N n − ( n ∈ N ) . Define matrices A n ∈ L ( C d n , C d n ) by A n = C (exp( − a ( N n − + 1) α ) , . . . , exp( − a ( N n ) α )) . Then s k ( A n ) = exp( − a ( N n − + k ) α ) (1 ≤ k ≤ d n )and | λ k ( A n ) | = exp( − ap αn ) (1 ≤ k ≤ d n ) , where p n := 1 d n N n X l = N n − +1 l α . Put H := ∞ M n =1 C d n , and let A : H → H be the block-diagonal operator( Ax ) n = A n x n . Clearly, the singular numbers of A are given by s k ( A ) = exp( − ak α ) and the moduliof the eigenvalues are the numbers exp( − ap αn ) occurring with multiplicity d n .Before checking that A has the desired properties we observe that(2) p αn = 1 d n N n X l = N n − +1 l α ≤ d n Z N n +1 N n − +1 x α == 1 α + 1 1 d n (( N n + 1) α +1 − ( N n − + 1) α +1 ) = 1 α + 1 N αn δ n , with lim n →∞ δ n = 1. The latter follows from the fact that the sequence N n waschosen to be super-exponentially increasing.Suppose now that b > a/ ( α + 1). Since | λ N n ( A ) | = exp( − ap αn ) we have | λ N n ( A ) | exp( bN αn ) ≥ exp( − aα + 1 N αn δ n + bN αn ) = exp( N αn ( b − aα + 1 δ n )) . Thus | λ N n ( A ) | exp( bN αn ) → + ∞ as n → ∞ , which means that λ ( A )
6∈ E ( b, α ). (cid:3) Remark 2.11.
Similar results have been obtained by K¨onig and Richter [KR,Proposition 1], though without estimates on the gauge of λ ( A ). ESOLVENT ESTIMATES 9 Resolvent estimates
In this section we shall derive an upper bound for the norm of the resolvent( zI − A ) − of A ∈ E ( a, α ) in terms of the distance of z to the spectrum of A and the departure from normality of A , a number quantifying the non-normalityof A . We shall employ a technique originally due to Henrici [Hen], who used it ina finite-dimensional context. The basic idea is to write A as a perturbation of anormal operator having the same spectrum as A by a quasi-nilpotent operator. Asimilar argument can be used to derive resolvent estimates for operators belongingto Schatten classes (see [Gil] (and references therein) and [Ban]).Following the idea outlined above we start with bounds for quasi-nilpotent op-erators. Proposition 3.1.
Let a, α > . (i) If A ∈ E ( a, α ; H, H ) is quasi-nilpotent, that is, σ ( A ) = { } , then (3) (cid:13)(cid:13) ( I − A ) − (cid:13)(cid:13) ≤ f a,α ( | A | a,α ) , where f a,α : R +0 → R +0 is defined by f a,α ( r ) = ∞ Y n =1 (1 + r exp( − an α )) . Moreover, f a,α has the following asymptoticss: (4) log f a,α ( r ) ∼ a − /α α α (log r ) /α as r → ∞ . (ii) If H is infinite-dimensional the estimate (3) is sharp in the sense that thereis a quasi-nilpotent B ∈ E ( a, α ; H, H ) such that (5) log (cid:13)(cid:13) ( I − zB ) − (cid:13)(cid:13) ∼ log f a,α ( | zB | a,α ) as | z | → ∞ . Proof.
Fix a, α > A is trace class and quasi-nilpotent a standard estimate (see, for exam-ple, [GGK, Chapter X, Theorem 1.1]) shows that (cid:13)(cid:13) ( I − A ) − (cid:13)(cid:13) ≤ ∞ Y n =1 (1 + s n ( A )) . Thus (cid:13)(cid:13) ( I − A ) − (cid:13)(cid:13) ≤ ∞ Y n =1 (1 + | A | a,α exp( − an α )) = f a,α ( | A | a,α ) . It remains to prove the growth estimate (4). We proceed by noting that f a,α extendsto an entire function of genus zero with f a,α (0) = 1. Moreover, the maximummodulus of f a,α ( z ) for | z | = r equals f a,α ( r ). The growth of f a,α can thus beestimated by(6) N ( r ) ≤ log f a,α ( r ) ≤ N ( r ) + Q ( r ) , where N ( r ) = R r t − n ( t ) dt , Q ( r ) = r R ∞ r t − n ( t ) dt and n ( r ) denotes the numberof zeros of f a,α lying in the closed disk with radius r centred at 0 (see [Boa, p. 47]). Since n ( r ) = ⌊ a − /α (log + r ) /α ⌋ , where log + ( r ) = max { , log r } and ⌊·⌋ denotesthe floor-function, we have, for r ≥ N ( r ) = a − /α Z r t − (log t ) /α dt + O (log r )= a − /α α α (log r ) /α + O (log r ) ;(7)while Q satisfies(8) Q ( r ) = O ((log r ) /α ) as r → ∞ . To see this, note that(9) Q ( r ) ≤ a − /α r Z ∞ r t − (log t ) /α dt = a − /α r Z ∞ log r e − u u /α du ;putting r = e s it thus suffices to show that e s Z ∞ s e − u u /α du = O ( s /α ) as s → ∞ . This, however, is the case since s − /α e s Z ∞ s e − u u /α du = Z ∞ s e − ( u − s ) ( u/s ) /α du = Z ∞ e − t (1 + t/s ) /α dt → s → ∞ . Combining (8), (7) and (6) the growth estimate (4) follows.(ii) Since H is infinite-dimensional, we may choose an orthonormal basis { h n } n ∈ N .Define the operator B ∈ L ( H, H ) by Bh n := exp( − an α ) h n +1 ( n ∈ N ) . It is not difficult to see that s n ( B ) = exp( − an α ) for n ∈ N , so that B ∈ E ( a, α ; H, H ).Before we proceed let c n := n X k =1 k α ( n ∈ N ) , and note that since R n x α dx ≤ P nk =1 k α ≤ R n +11 x α dx , we have1 α + 1 n α +1 ≤ c n ≤ α + 1 ( n + 1) α +1 ( n ∈ N ) . The operator B is quasi-nilpotent, since k B n k = exp( − ac n ) ≤ exp( − aα + 1 n α +1 ) , which implies k B n k /n → n → ∞ .It order to determine the asymptotics of log (cid:13)(cid:13) ( I − zB ) − (cid:13)(cid:13) we start by notingthat(10) (cid:13)(cid:13) ( I − zB ) − (cid:13)(cid:13) ≥ (cid:13)(cid:13) ( I − zB ) − h (cid:13)(cid:13) = k ∞ X n =0 ( zB ) n h k = ∞ X n =0 | z | n exp( − ac n ) ≥ ∞ X n =0 | z | n exp( − aα + 1 ( n + 1) α +1 ) ≥ | z | − g ( | z | ) , where g ( r ) := ∞ X n =1 r n exp( − aα + 1 n α +1 ) ( r ∈ R +0 ) . ESOLVENT ESTIMATES 11
Thus(11) 2 log f a,α ( | zB | a,α ) ≥ (cid:13)(cid:13) ( I − zB ) − (cid:13)(cid:13) ≥ − | z | + log g ( | z | ) , which shows that in order to obtain the desired asymptotics (5) it suffices to provethat(12) log g ( r ) ∼ a − /α αα + 1 (log r ) /α as r → ∞ . In order to establish the asymptotics above we introduce the maximum term(13) µ ( r ) := max ≤ n< ∞ r n exp( − aα + 1 n α +1 ) ( r ∈ R +0 ) . Since g extends to an entire function of finite order we have (see, for example, [PS,Problem 54]) log µ ( r ) ∼ log g ( r ) as r → ∞ , which implies that it now suffices to show that µ has the desired asymptotics(14) log µ ( r ) ∼ a − /α αα + 1 (log r ) /α as r → ∞ . We now estimate µ ( r ) for fixed r . Define the function m r : R +0 → R +0 by m r ( x ) = exp( − aα + 1 x α +1 + 2 x log r ) . It turns out that m r has a maximum at x r = a − /α (log r ) /α and that m r ismonotonically increasing on (0 , x r ) and monotonically decreasing on ( x r , ∞ ). Thus(15) log m r ( x r − ≤ log µ ( r ) ≤ log m r ( x r ) . Write x r − δ r x r and note that δ r → r → ∞ . Observing thatlog m r ( x r )(log r ) /α = 2 a − /α αα + 1whilelog m r ( x r − r ) /α = log m r ( δ r x r )(log r ) /α = 2 a − /α (cid:18) − δ α +1 r α + 1 + δ r (cid:19) → a − /α αα + 1as r → ∞ we conclude, using (15), that (14) holds. This implies (12), which yields(5), as required. (cid:3) Remark 3.2. (i) The bound for the growth of the resolvent of a quasi-nilpotent A ∈ E ( a, α )given in the above proposition is an improvement compared to those obtainablefrom the usual estimates for operators in the Schatten classes. Indeed, if A belongsto the Schatten p -class (i.e. s ( A ) is p -summable) for some p >
0, then (cid:13)(cid:13) ( I − A ) − (cid:13)(cid:13) ≤ f p ( k A k p )where k A k p denotes the Schatten p -(quasi) norm of A . Here, f p : R +0 → R +0 isgiven by f p ( r ) = exp( a p r p + b p ) , where a p and b p are positive numbers depending on p , but not on A (see, forexample, [Sim1] or [Ban, Theorem 2.1], where a discussion of the constants a p and b p can be found). (ii) Closer inspection of the proof yields the following explicit upper bound for f a,α log f a,α ( r ) ≤ a − /α (cid:18) α α (log + r ) /α + r Γ(1 + 1 /α, log + r ) (cid:19) , where Γ( β, s ) denotes the incomplete gamma functionΓ( β, s ) = Z ∞ s exp( − t ) t β − dt . This follows from (6) together with the estimate n ( r ) ≤ a − /α (log + r ) /α .A simple consequence of the proposition is the following estimate for the growthof the resolvent of a quasi-nilpotent A ∈ E ( a, α ). Corollary 3.3. If A ∈ E ( a, α ) is quasi-nilpotent, then (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ | z | − f a,α ( | z | − | A | a,α ) for z = 0 . The proposition above can be used to obtain growth estimates for the resolventsof any A ∈ E ( a, α ) by means of the following device. Theorem 3.4.
Let A ∈ S ∞ . Then A can be written as a sum A = D + N, such that (i) D ∈ S ∞ , N ∈ S ∞ ; (ii) D is normal and λ ( D ) = λ ( A ) ; (iii) N and ( zI − D ) − N are quasi-nilpotent for every z ∈ ̺ ( D ) = ̺ ( A ) .Proof. See [Ban, Theorem 3.2]. (cid:3)
This result motivates the following definition.
Definition 3.5.
Let A ∈ S ∞ . A decomposition A = D + N with D and N enjoying properties (i–iii) of the previous theorem is called a Schurdecomposition of A . The operators D and N will be referred to as the normal andthe quasi-nilpotent part of the Schur decomposition of A , respectively. Remark 3.6. (i) The terminology stems from the fact that in the finite dimensional settingthe decomposition in Theorem 3.4 can be obtained as follows: since any matrix isunitarily equivalent to an upper-triangular matrix by a classical result due to Schur,it suffices to establish the result for matrices of this form. In this case, simply choose D to be the diagonal part, and N the off-diagonal part of the matrix.(ii) The decomposition is not unique, not even modulo unitary equivalence: thereis a matrix A with two Schur decompositions A = D + N and A = D + N suchthat N is not unitarily equivalent to N (see [Ban, Remark 3.5 (i)]). Note, however,that the normal parts of any two Schur decompositions of a given compact operatorare always unitarily equivalent.Using the results in the previous section we are able to locate the position of thenormal part and the quasi-nilpotent part of an operator A ∈ E ( a, α ) in the scale ofexponential classes. ESOLVENT ESTIMATES 13
Proposition 3.7.
Let A ∈ E ( a, α ) . If A = D + N is a Schur decomposition of A with normal part D and quasi-nilpotent part N , then (i) D ∈ E ( a/ (1 + α ) , α ) with | D | a/ (1+ α ) ,α ≤ | A | a,α ; (ii) N ∈ E ( a ′ , α ) with | N | a ′ ,α ≤ | A | a,α , where a ′ = a (1 + (1 + α ) /α ) − α .Proof. Since D is normal, its singular numbers coincide with its eigenvalues, whichin turn coincide with the eigenvalues of A . Assertion (i) is thus a consequence ofProposition 2.10, while assertion (ii) follows from (i) and Proposition 2.8 by taking N = A − D . (cid:3) Remark 3.8.
Assertion (i) above is sharp in the following sense: there is A ∈ E ( a, α ) such that for any normal part D of A we have D E ( b, α ) whenever b > a/ (1 + α ). This follows from the corresponding statement in Proposition 2.10and the fact that all normal parts of A are unitarily equivalent.For later use we define the following quantities, originally introduced by Henrici[Hen]. Definition 3.9.
Let a, α > ν a,α : S ∞ → R +0 ∪ {∞} by ν a,α ( A ) := inf { | N | a,α : N is a quasi-nilpotent part of A } . We call ν a,α ( A ) the ( a, α ) -departure from normality of A . Remark 3.10.
Henrici originally introduced this quantity for matrices and withthe ( a, α )-gauge of N replaced by the Hilbert-Schmidt norm. For a discussion of thecase where the ( a, α )-gauge is replaced by a Schatten norm and its uses to obtainresolvent estimates for Schatten class operators see [Ban].The term ‘departure from normality’ is justified in view of the following charac-terisation. Proposition 3.11.
Let A ∈ E ( a, α ) . Then ν a,α ( A ) = 0 ⇔ A is normal . Proof.
Let ν a,α ( A ) = 0. Then there exists a sequence of Schur decompositions withquasi-nilpotent parts N n such that | N n | a,α →
0. Thus k A − D n k = k N n k = s ( N n ) ≤ exp( − a ) | N | a,α → , where D n are the corresponding normal parts. Thus A is a limit of normal operatorsconverging in the uniform operator topology and is therefore normal. The converseis trivial. (cid:3) For a given A ∈ E ( a, α ) the departure from normality is difficult to calculate.The following simple but somewhat crude bound is useful in practice. Proposition 3.12.
Let A ∈ E ( a, α ) . Then ν b,α ( A ) ≤ | A | a,α whenever b ≤ a (1 + (1 + α ) /α ) − α .Proof. Follows from Proposition 3.7 together with the fact that | N | b,α ≤ | N | a ′ ,α whenever b ≤ a ′ . (cid:3) We are now ready to deduce resolvent estimates for arbitrary A ∈ E ( a, α ). Usinga Schur decomposition A = D + N with D normal and N quasi-nilpotent we consider A as a perturbation of D by N . Since D is normal we have(16) (cid:13)(cid:13) ( zI − D ) − (cid:13)(cid:13) = 1 d ( z, σ ( D )) ( z ∈ ̺ ( D )) , where for z ∈ C and σ ⊂ C closed, d ( z, σ ) := inf λ ∈ σ | z − λ | denotes the distance of z to σ . The influence of the perturbation N on the otherhand, is controlled by Proposition 3.1. All in all, we have the following. Theorem 3.13.
Let A ∈ E ( a, α ) . If b ≤ a (1 + (1 + α ) /α ) − α , then (17) (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ d ( z, σ ( A )) f b,α (cid:18) ν b,α ( A ) d ( z, σ ( A )) (cid:19) . Proof.
Fix b ≤ a (1 + (1 + α ) /α ) − α . Then there is a Schur decomposition of A withnormal part D and quasi-nilpotent part N ∈ E ( b, α ) by Proposition 3.7. Sincethe bound above is trivial for z ∈ σ ( A ) we may assume z ∈ ̺ ( A ). As D and N stem from a Schur decomposition of A we see that ( zI − D ) − exists (because σ ( A ) = σ ( D )) and that ( zI − D ) − N is quasi-nilpotent. Moreover | ( zI − D ) − N | b,α ≤ (cid:13)(cid:13) ( zI − D ) − (cid:13)(cid:13) | N | b,α = | N | b,α d ( z, σ ( D )) , by (16) and Proposition 2.5. Thus ( I − ( zI − D ) − N ) is invertible in L and (cid:13)(cid:13) ( I − ( zI − D ) − N ) − (cid:13)(cid:13) ≤ f b,α (cid:18) | N | b,α d ( z, σ ( D )) (cid:19) , by Proposition 3.1. Now, since ( zI − A ) = ( zI − D )( I − ( zI − D ) − N ), we concludethat ( zI − A ) is invertible in L and (cid:13)(cid:13) ( z − A ) − (cid:13)(cid:13) ≤ (cid:13)(cid:13) ( I − ( zI − D ) − N ) − (cid:13)(cid:13) (cid:13)(cid:13) ( zI − D ) − (cid:13)(cid:13) ≤ d ( z, σ ( D )) f b,α (cid:18) | N | b,α d ( z, σ ( D )) (cid:19) . Taking the infimum over all Schur decompositions while using σ ( A ) = σ ( D ) onceagain the result follows. (cid:3) Remark 3.14. (i) The estimate remains valid if ν b,α ( A ) is replaced by something larger, forexample by the upper bound given in Proposition 3.12.(ii) The estimate is sharp in the sense that if A is normal then (17) reduces tothe sharp estimate (16).4. Bounds for the spectral distance
Using the resolvent estimates obtained in the previous section it is possible togive upper bounds for the Hausdorff distance of the spectra of operators in E ( a, α ).Recall that the Hausdorff distance
Hdist ( ., . ) is the following metric defined on thespace of compact subsets of C Hdist ( σ , σ ) := max { ˆ d ( σ , σ ) , ˆ d ( σ , σ ) } , ESOLVENT ESTIMATES 15 where ˆ d ( σ , σ ) := sup λ ∈ σ d ( λ, σ )and σ and σ are two compact subsets of C . For A, B ∈ L we borrow terminologyfrom matrix perturbation theory and call ˆ d ( σ ( A ) , σ ( B )) the spectral variation of A with respect to B and Hdist ( σ (A) , σ (B)) the spectral distance of A and B .The main tool to bound the spectral variation is the following result, which isbased on a simple but powerful argument usually credited to Bauer and Fike [BauF]who first employed it in a finite-dimensional context. Proposition 4.1.
Let A ∈ S ∞ . Suppose that there is a strictly monotonicallyincreasing surjection g : R +0 → R +0 such that (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ g ( d ( z, σ ( A )) − ) ( ∀ z ∈ ̺ ( A )) . Then for any B ∈ L , the spectral variation of B with respect to A satisfies ˆ d ( σ ( B ) , σ ( A )) ≤ h ( k A − B k ) , where h : R +0 → R +0 is the function defined by h ( r ) = (˜ g ( r − )) − and ˜ g : R +0 → R +0 is the inverse of the function g .Proof. Let B ∈ L . In what follows we shall use the abbreviations d := d ( z, σ ( A )) , E := B − A. Without loss of generality we may assume that E = 0. We shall first establish thefollowing implication:(18) z ∈ σ ( B ) ∩ ̺ ( A ) ⇒ k E k − ≤ (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) . To see this let z ∈ σ ( B ) ∩ ̺ ( A ) and suppose to the contrary that (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) k E k < . Then (cid:0) I − ( zI − A ) − E (cid:1) is invertible in L , so ( zI − B ) = ( zI − A ) (cid:0) I − ( zI − A ) − E (cid:1) isinvertible in L . Hence z ∈ ̺ ( B ) which contradicts z ∈ σ ( B ). Thus the implication(18) holds.In order to prove the proposition it suffices to show that(19) z ∈ σ ( B ) ⇒ d ( z, σ ( A )) ≤ h ( k E k ) , which is proved as follows. Let z ∈ σ ( B ). If z ∈ σ ( A ) there is nothing to prove.We may thus assume that z ∈ ̺ ( A ). Hence, by (18), k E k − ≤ (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ g ( d − ) . Since g is strictly monotonically increasing, so is ˜ g . Thus˜ g ( k E k − ) ≤ d − , and hence d ( z, σ ( A )) = d ≤ (˜ g ( k E k − )) − = h ( k E k ) = h ( k A − B k ) . (cid:3) Using the proposition above together with the resolvent estimates in Theo-rem 3.13 we now obtain the following spectral variation and spectral distance for-mulae.
Theorem 4.2.
Let A ∈ E ( a, α ) and define a ′ := a (1 + (1 + α ) /α ) − α . (i) If B ∈ L and b > a ′ then (20) ˆ d ( σ ( B ) , σ ( A )) ≤ ν ( A ) b,α h b,α (cid:18) k A − B k ν b,α ( A ) (cid:19) . (ii) If B ∈ E ( a, α ) and b > a ′ then (21) Hdist ( σ ( A ) , σ ( B )) ≤ mh b,α (cid:18) k A − B k m (cid:19) , where m := max { ν ( A ) b,α , ν ( B ) b,α } .Here, the function h b,α : R +0 → R +0 is given by h b,α ( r ) := (˜ g b,α ( r − )) − , where ˜ g : R +0 → R +0 is the inverse of the function g b,α : R +0 → R +0 defined by g b,α ( r ) := rf b,α ( r ) , and f a,α is the function defined in Proposition 3.1.Proof. To prove (i) fix b ≥ a ′ . The assertion now follows from the previous propo-sition by noting that (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ ν b,α ( A ) g b,α (cid:18) ν b,α ( A ) d ( z, σ ( A )) (cid:19) by Theorem 3.13. To prove (ii) fix b ≥ a ′ . Then it is not difficult see that (cid:13)(cid:13) ( zI − A ) − (cid:13)(cid:13) ≤ m g b,α (cid:18) md ( z, σ ( A )) (cid:19) , and (cid:13)(cid:13) ( zI − B ) − (cid:13)(cid:13) ≤ m g b,α (cid:18) md ( z, σ ( B )) (cid:19) , and the assertion follows as in the proof of (i). (cid:3) Remark 4.3. (i) Note that lim r ↓ h b,α ( r ) = 0, so the estimate for the spectral distance becomessmall when k A − B k is small. In fact, it can be shown thatlog h b,α ( r ) ∼ − b / (1+ α ) (cid:18) αα (cid:19) α/ (1+ α ) | log r | α/ (1+ α ) as r ↓ . This follows from the asymptotics in Proposition 3.1 together with the fact that iflog f ( r ) ∼ a (log r ) β , then log ˜ f ( r ) ∼ a − /β (log r ) /β where ˜ f is the inverse of f .(ii) It is not difficult to see, for example by arguing as in the proof of part (ii)of the theorem, that the inequalities (20) and (21) above remain valid if ν b,α ( A ) or ν b,α ( B ) is replaced by something larger — for example, by the upper bounds givenin Proposition 3.12.(iii) Assertion (ii) of the theorem is sharp in the sense that if both operators arenormal, then (ii) reduces toHdist ( σ ( A ) , σ ( B )) ≤ k A − B k . ESOLVENT ESTIMATES 17 Appendix: Monotone arrangements
In this appendix we present a number of results concerning sequences and theirarrangements used in Section 2.Let a : N → R be a sequence. Define k a k + := sup n ∈ N a n , k a k − := inf n ∈ N a n . For u : N → R call rank ( u ) := card { n ∈ N : u n = 0 } . Definition 5.1.
Let a : N → R be a sequence. Let extended real-valued sequences a (+) and a ( − ) be defined by a (+) : N → R ∪ {−∞ , ∞} , a (+) n := inf (cid:8) k a − u k + : rank u < n (cid:9) ,a ( − ) : N → R ∪ {−∞ , ∞} , a ( − ) n := sup (cid:8) k a − u k − : rank u < n (cid:9) . We call a (+) the decreasing arrangement of a , and a ( − ) the increasing arrangementof a .This terminology is justified in view of the fact that a (+) (respectively a ( − ) ) isa decreasing (respectively increasing) sequence. Moreover, if a is monotonicallydecreasing, then a (+) = a , and similarly for monotonically increasing sequences.More generally, we will consider monotone arrangements of collections of se-quences by first amalgamating them into one sequence and then regarding theresulting monotone arrangement. A more precise definition is the following. Definition 5.2.
Given K real-valued sequences { a (1) n } n ∈ N , . . . , { a ( K ) n } n ∈ N , define anew sequence a : N → R by a ( k − K + i = a ( i ) k for k ∈ N and 1 ≤ i ≤ K .
We then call a (+) ( a ( − ) ) the decreasing (increasing) arrangement of the K sequences a (1) , · · · , a ( K ) . Our application of decreasing arrangements will typically be to singular numbersequences, all of which converge to zero at some stretched exponential rate. Techni-cally and notationally it is preferable to work with the logarithms of reciprocals ofsuch sequences, that is, increasing sequences converging to + ∞ at some polynomialrate.The following is the main result of this appendix. Proposition 5.3.
Let α > and K ∈ N . Suppose that for each k ∈ { , . . . , K } weare given a real sequence a ( k ) , a positive constant a k > , and a real number A k .Let a ( − ) denote the increasing arrangement of the K sequences a (1) , . . . , a ( K ) , anddefine c = K X k =1 a − /αk ! − α . (i) If a ( k ) n ≥ a k n α + A k ( ∀ n ∈ N , k ∈ { , . . . , K } ) , then a ( − ) n ≥ cn α + min { A , . . . , A K } ( ∀ n ∈ N ) . (ii) If a ( k ) n ≤ a k n α + A k ( ∀ n ∈ N , k ∈ { , . . . , K } ) , then a ( − ) n ≤ c ( n + K ) α + max { A , . . . , A K } ( ∀ n ∈ N ) . Proof.
For k ∈ { , . . . , K } set a ( k )0 = −∞ and, for r ∈ R , define the countingfunctions µ k ( r ) := card n n ∈ N : a ( k ) n ≤ r o µ ( r ) := card n n ∈ N : a ( − ) n ≤ r o . The following relations are easily verified. We have(22) µ ( r ) = K X k =1 µ k ( r ) , and a ( − ) µ ( r ) ≤ r ( ∀ r ∈ R ) , (23) µ ( a ( − ) n ) ≥ n ( ∀ n ∈ N ) . (24)(i) Set C = min { A , . . . , A K } . Since for each k ∈ { , . . . , K } , n n ∈ N : a ( k ) n ≤ r o ⊂ { n ∈ N : a k n α + C ≤ r } , we have, for r ≥ C , µ k ( r ) ≤ card { n ∈ N : a k n α + C ≤ r } = $(cid:18) r − Ca k (cid:19) /α % , where ⌊·⌋ denotes the floor function. Thus, using (22), we have(25) µ ( r ) = K X k =1 µ k ( r ) ≤ ( r − C ) /α K X k =1 a − /αk . If n ∈ N then a ( − ) n ≥ C , so combining (25) with (24) gives( a ( − ) n − C ) /α K X k =1 a − /αk ≥ µ ( a ( − ) n ) ≥ n , from which (i) follows.(ii) Set C = max { A , . . . , A K } . Since for each k ∈ { , . . . , K } , { n ∈ N : a k n α + C ≤ r } ⊂ n n ∈ N : a ( k ) n ≤ r o , we have, for r ≥ C , $(cid:18) r − Ca k (cid:19) /α % = card { n ∈ N : a k n α + C ≤ r } ≤ µ k ( r ) . Thus, using (22), we have(26) µ ( r ) = K X k =1 µ k ( r ) ≥ ( r − C ) /α K X k =1 a − /αk ! − K .
ESOLVENT ESTIMATES 19
Now fix n ∈ N . Choose r ≥ C such that(27) n = ( r − C ) /α K X k =1 a − /αk ! − K .
From (26) and (27) we see that n ≤ µ ( r ). Using (23), together with the fact that n a ( − ) n is monotonically increasing, now yields a ( − ) n ≤ a ( − ) µ ( r ) ≤ r = c ( n + K ) α + C .
Since n was arbitrary, (ii) follows. (cid:3) Corollary 5.4.
Let α > and K ∈ N . Suppose that for each k ∈ { , . . . , K } weare given a real sequence b ( k ) , a positive constant a k > , and a real number B k .Let b (+) denote the decreasing arrangement of the K sequences b (1) , . . . , b ( K ) anddefine c := K X k =1 a − /αk ! − α . (i) If b ( k ) n ≤ B k exp( − a k n α ) ( ∀ n ∈ N , k ∈ { , . . . , K } ) , then b (+) n ≤ B exp( − cn α ) ( ∀ n ∈ N ) , where B = max { B , . . . , B K } . (ii) If b ( k ) n ≥ B k exp( − a k n α ) ( ∀ n ∈ N , k ∈ { , . . . , K } ) , then b (+) n ≥ B exp( − c ( n + K ) α ) ( ∀ n ∈ N ) , where B = min { B , . . . , B K } . Acknowledgements.
I would like to thank Oliver Jenkinson for the many discus-sions that helped to shape this article, in particular the results in Section 2 andin the Appendix. The research described in this article was partly supported byEPSRC Grant GR/R64650/01.
References [Ban] OF Bandtlow (2004) Estimates for norms of resolvents and an application to the per-turbation of spectra;
Math. Nachr. , 3–11[BanC] OF Bandtlow and C-H Chu (2008) Eigenvalue decay of operators on harmonic functionspaces; preprint [BanJ1] OF Bandtlow and O Jenkinson (2007) Explicit a priori bounds on transfer operatoreigenvalues;
Commun. Math. Phys. , 901–905[BanJ2] OF Bandtlow and O Jenkinson (2007) On the Ruelle eigenvalue sequence; to appear in
Ergod. Th. & Dynam. Sys. [BanJ3] OF Bandtlow and O Jenkinson (2008) Explicit eigenvalue estimates for transfer operatorsacting on spaces of holomorphic functions;
Adv. Math. , 902–925[BauF] FL Bauer and CT Fike (1960) Norms and exclusion theorems;
Num. Math. , 42–53[BerF] G Bery and G Forst (1975) Potential Theory on Locally Compact Groups ; Heidelberg,Springer[Boa] RP Boas (1954)
Entire Functions ; New York, Academic Press[Cal] JW Calkin (1941) Two sided ideals and congruences in the ring of bounded operatorsin Hilbert space;
Ann. Math. , 839–873[DS] N Dunford, JT Schwartz (1963) Linear Operators, Vol. 2 ; New York, Interscience [Gil] MI Gil’ (2003)
Operator Functions and Localization of Spectra ; Berlin, Springer[GGK] I Gohberg, S Goldberg, MA Kaashoek (1990)
Classes of Linear Operators Vol. 1 ; Basel,Birkh¨auser[GK] I Gohberg, MG Krein (1969)
Introduction to the Theory of Linear Non-Selfadjoint Op-erators ; Providence, AMS[Hen] P Henrici (1962) Bounds for iterates, inverses, spectral variation and fields of values ofnon-normal matrices;
Num. Math. , 24–40[KR] H K¨onig and S Richter (1984) Eigenvalues of integral operators defined by analytickernels; Math. Nachr. , 141–155[Nel] E Nelimarkka (1982) On λ ( P, N )-nuclearity and operator ideals;
Math. Nachr. , 231–237[Pie] A Pietsch (1988) Eigenvalues and s -Numbers ; Cambridge, CUP[Pok] A Pokrzywa (1985) On continuity of spectra in norm ideals; Lin. Alg. Appl. , 121–130[PS] G P´olya and G Szeg¨o (1976) Problems and Theorems in Analysis, Volume 2 ; Berlin,Springer[Rin] JR Ringrose (1971)
Compact Non-Self-Adjoint Operators ; London, van Nostrand[Rue] D Ruelle (2004)
Thermodynamic formalism: the mathematical structures of equilibriumstatistical mechanics ; Cambridge CUP[Sim1] B Simon (1977) Notes on infinite determinants of Hilbert space operators;
Adv. in Math. , , 244–273[Sim2] B Simon (1979) Trace ideals and their applications ; Cambridge, CUP
School of Mathematical Sciences, Queen Mary, University of London, London E34NS, UK
E-mail address ::