Abstract: We study the region of complete localization in a class of random operators which includes random Schrodinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity.

Abstract: We develop a dilation theory on noncommutative varieties determined by row contractions T: = [T 1 ,..., T n ] subject to constraints such as p(T 1 ,...,T n ) = 0, p ∈ P, where T is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B 1 ,...,B n ] satisfying the same constraints as T, which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S 1 ,..., S n ] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C*-algebra generated by B 1 ,...,B n and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function Θ T is associated with any (constrained) row contraction T and it is proved that I - Θ T Θ* T = K T K* T , where K T is the (constrained) Poisson kernel of T. Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.

Abstract: The local L 2-mapping property of Fourier integral operators has been established in Hormander (1971) and in Eskin (1970). In this article, we treat the global L 2-boundedness for a class of operators that appears naturally in many problems. As a consequence, we improve known global results for several classes of pseudodifferential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara (1978) or Kumano-go (1976). As an application, we show a global smoothing estimate for generalized Schrodinger equations which extends the results of Ben-Artzi and Devinatz (1991) and Walther (1999); (2002).

Abstract: In this article the p-adic Lizorkin spaces of test functions and distributions are introduced. Multi-dimensional Vladimirov’s and Taibleson’s fractional operators, and a class of p-adic pseudo-differential operators are studied on these spaces. Since the p-adic Lizorkin spaces are invariant under these operators, they can play a key role in considerations related to fractional operator problems. Solutions of pseudo-differential equations are also constructed. Some problems of spectral analysis of pseudo-differential operators are studied. p-Adic multidimensional Tauberian theorems connected with these pseudo-differential operators for the Lizorkin distributions are proved.

Abstract: We prove comparison theorems for the H ∞-calculus that allow to transfer the property of having a bounded H ∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L p -scale. We apply the results to give new proofs on the H ∞-calculus for elliptic differential operators, including Schrodinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ∂Ω ∈ C 1,1 has a bounded H ∞-calculus in the Helmholtz scale L p,σ (Ω), p ∈ (1,∞).

Abstract: Two generalizations of the notion of principal eigenvalue for elliptic operators in $\R^N$ are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and ``limit periodic'' operators. These results apply to questions of existence and uniqueness for some semi-linear problems in all of space. We also indicate several outstanding open problems and formulate some conjectures.

Abstract: We consider a generalization of isometric Hilbert space operators to the multivariable setting. We study some of the basic properties of these tuples of commuting operators and we explore several examples. In particular, we show that the d-shift, which is important in the dilation theory of d-contractions (or row contractions), is a d-isometry. As an application of our techniques we prove a theorem about cyclic vectors in certain spaces of analytic functions that are properly contained in the Hardy space of the unit ball of \( \mathbb{C}^d \).

Abstract: Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*-modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*-modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*-modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*-modules leaves a frame or a non-frame set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*-modules may possess infinitely many alternative duals due to the existence of zero-divisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert $C^*$-modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*-algebra is a commutative W*-algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also prove the existence and uniqueness of the best Parseval multi-frame approximations for multi-frame generators of unitary groups on Hilbert C*-modules when the underlying C*-algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*-module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*-modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*-module. For the perturbation of frames and Riesz bases in Hilbert C*-modules we prove that the Casazza-Christensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*-modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a…

Abstract: We present some properties of mixture and generalized mixture operators, with special stress on their monotonicity. We introduce new sufficient conditions for weighting functions to ensure the monotonicity of the corresponding operators. However, mixture operators, generalized mixture operators neither quasi-arithmetic means weighted by a weighting function need not be non- decreasing operators, in general.

Abstract: The Hankel transform transplantation operator is investigated by means of a suitably established local version of the Calderon-Zygmund operator theory. This approach produces weighted norm inequalities with weights more general than previously considered power weights. Moreover, it also allows to obtain weighted weak type $(1,1)$ inequalities, which seem to be new even in the unweighted setting. As a typical application of the transplantation, multiplier results in weighted $L^p$ spaces with general weights are obtained for the Hankel transform of any order $\alpha > -1$ greater than $-1$ by transplanting cosine transform multiplier results.

Abstract: We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at x ~ ∞ and we give a canonical representation of σess(H) in terms of spectra of limits at infinity of translations of H.

Abstract: We study spectra of Schrodinger operators on ℝ d . First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values μ n of the difference of the semigroups as n→∞ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrodinger operators. The single site potential u is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Holder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Holder continuity of the integrated density of states.

Abstract: A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of A n , n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A* has trivial kernel.

Abstract: We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro’s theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro’s theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras.

Abstract: This survey is devoted to the history and the current state of the theory of regularized traces of linear operators. The main focus is on operators with discrete spectrum. Several appendices are devoted to closely related areas of spectral theory and to the formulation of some important unsolved problems.

Abstract: We find a sharp combinatorial bound for the metric entropy of sets in R n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central Limit Theorem if the square root of its combinatorial dimension is integrable. 2. The uniform entropy is equivalent to the combinatorial dimension under minimal regularity. Our method also constructs a nicely bounded coordinate section of a symmetric convex body in R n . In the operator theory, this essentially proves for all normed spaces the restricted invertibility principle of Bourgain and Tzafriri.

Abstract: In this article we study approximation properties for the class of general integral operators of the form where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of closed subsets of G with suitable properties and, for every w>0, μ Hw is a regular measure on Hw We give pointwise, uniform and modular convergence theorems in abstract modular spaces and we apply the results to some kind of discrete operators including the sampling-type series.