Resolvent Estimates for Operators Belonging to Exponential Classes

Abstract

Abstract. For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that sn(A) = O(exp(−an α )), where sn(A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A) −1 of A in terms of a quantity describing the departure 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 the spectra of two operators in E(a, α).

Rigorous dimension estimates for Cantor sets arising in Zaremba theory

“…2]), so that L A,s belongs to an exponential class (cf. [1,2]) and is in particular a trace class operator, from which the existence and above properties of trace and determinant follow (see [27]).…”

Explicit upper bounds for the spectral distance of two trace class operators

“…In Section 4 we combine the upper and lower bounds to obtain our main result. We finish (Section 5) by comparing our bound to Elsner's and show that it reduces to or improves the bounds in [1,2].…”

“…This leads to the notion of exponential classes introduced in [2]. Naturally occurring operators belonging to these classes include integral operators with analytic kernels (see [13]), composition operators with strictly contracting holomorphic symbols, or, more generally, transfer operators arising from real analytic expanding maps, which play an important role in smooth ergodic theory (see [4]).…”

“…Using this approach upper bounds for the spectral distance of Schatten class operators have been obtained by Gil' in a series of papers (see [8] and references therein for an overview) and one of the authors (see [1,2]). Similar bounds, less sharp, but valid more generally for operators in symmetrically normed ideals are due to Pokrzywa [17], using the same method.…”

Spectral structure of transfer operators for expanding circle maps

“…Thus J and hence L are trace-class. In fact, using results from [2] it is possible to show that both the singular values and the eigenvalues of L decay at an exponential rate, a property that L shares with other transfer operators arising from analytic maps (see, for example, [9,3]). …”