Let $E$ be a complex Banach space, with the unit ball $B_E$. We study the spectrum of a bounded weighted composition operator $u{C}_{\varphi }$ on $H^{\infty }\left(B_E\right)$ determined by an analytic symbol $\varphi$ with a fixed point in $B_E$ such that $\varphi \left(B_E\right)$ is a relatively compact subset of $E$, where $u$ is an analytic function on $B_E$.

We describe the spectra of Jacobi operators J with some irregular entries. We divide ℝ into three “spectral regions” for J and using the subordinacy method and asymptotic methods based on some particular discrete versions of Levinson’s theorem we prove the absolute continuity in the first region and the pure pointness in the second. In the third region no information is given by the above methods, and we call it the “uncertainty region”. As an illustration, we introduce and analyse the OP family...

We describe how the resolution of a kernel-based interpolation problem can be associated with a spectral problem. An integral operator is defined from the embedding of the considered Hilbert subspace into an auxiliary Hilbert space of square-integrable functions. We finally obtain a spectral representation of the interpolating elements which allows their approximation by spectral truncation. As an illustration, we show how this approach can be used to enforce boundary conditions in kernel-based...

Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${}^N$ or on the Segal-Bargmann space over ${ℂ}^N$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression $A_n$ of A to the linear span of the monomials $z_1^{k_1}...z_N^{k_N}:0\le k_j\le n$. Unfortunately, in general the spectrum of $A_n$ does not mimic the spectrum of A as...

The iteration subspace method for approximating a few points of the spectrum of a positive linear bounded operator is studied. The behaviour of eigenvalues and eigenvectors of the operators $A_n$ arising by this method and their dependence on the initial subspace are described. An application of the Schmidt orthogonalization process for approximate computation of eigenelements of operators $A_n$ is also considered.

We develop a spectral-theoretic harmonic analysis for an arbitrary UMD space X. Our approach utilizes the spectral decomposability of X and the multiplier theory for $L_X^p$ to provide on the space X itself analogues of the classical themes embodied in the Littlewood-Paley Theorem, the Strong Marcinkiewicz Multiplier Theorem, and the M. Riesz Property. In particular, it is shown by spectral integration that classical Marcinkiewicz multipliers have associated transforms acting on X.

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ\left(U\right)}_{n=1}^{\infty }$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ\left(U\right)=1/n{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))U^k$. We show that this sequence ${ₙ\left(U\right)}_{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

This paper gives a survey of some recent developments in the spectral theory of linear operators on Banach spaces in which the Hilbert transform and its abstract analogues play a fundamental role.

Given an orthogonal projection P and a free unitary Brownian motion $Y={\left(Yₜ\right)}_{t\ge 0}$ in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...

In this survey, we summarise some of the recent progress on the structure of spectral isometries between C*-algebras.