Advances in Operator Theory最新文献

A linear bounded operator T on a complex Banach space X is said to be power-regular if the sequence ({Vert T^n xVert ^{frac{1}{n}}}_{n=1}^{infty }) is convergent for every (xin X). For unilateral weighted shift S, we give a sufficient condition that S is power-regular. As an application, we construct a class of power-regular operators. Moreover, we show that there exist invertible power-regular bilateral weighted shifts, whose inverses are not power-regular.

In this paper, we investigate power-bounded operators, including surjective isometries, on Banach spaces. Koehler and Rosenthal asserted that an isolated point in the spectrum of a surjective isometry on a Banach space lies in the point spectrum, with the corresponding eigenspace having an invariant complement. However, they did not provide a detailed proof of this claim, at least as understood by the authors of this manuscript. Here, by applications of a theorem of Gelfand and the Riesz projections, we demonstrate that the theorem of Koehler and Rosenthal holds for any power-bounded operator on a Banach space. This not only furnishes a detailed proof of the theorem but also slightly generalizes its scope. As a result, we establish that if (T: X rightarrow X) is a power-bounded operator on a Banach space X whose spectrum consists of finitely many points ({lambda _1, lambda _2, dots , lambda _m}), then for every (1 le i, j le m), there exist projections (P_j) on X such that (P_iP_j=delta _{ij}P_i), (sum _{j=1}^mP_j=I), and (T=Sigma _{j=1}^m lambda _j P_j). It follows that such an operator T is an algebraic operator.

Given a smooth manifold M (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential calculus on manifolds (with or without boundary) developed in Ruzhansky and Tokmagambetov (Int Math Res Not IMRN 12:3548–3615, 2016), the Calderón–Vaillancourt Theorem and the global functional calculus in Cardona et al. (Adv Oper Theory arXiv:2101.02519, 2020), we determine the singularity orders in the regularisation of traces and the sharp regularity orders for the Dixmier traceability of the global Hörmander classes. Our analysis (free of coordinate systems) allows us to obtain non-harmonic analogues of several classical results arising from the microlocal analysis of regularised traces for pseudo-differential operators with symbols defined by localisations.

This paper concerns the spectral structure of hypercyclic and supercyclic operators defined on Banach spaces, or defined on Hilbert spaces. We also consider the spectral properties of operators in Hilbert spaces that commute with a hypercyclic operator. A result of Herrero and Kitai (Proc Am Math Soc 116(3):873–875, 1992) is extended to Drazin invertible operators. In particular, a Drazin invertible operator is hypercyclic if and only if is invertible. An analogous result holds for supercyclic operators T in the case were the dual (T^*) has empty point spectrum.

Quantum channels are usually studied as the completely positive trace preserving linear mapping of the space of normal quantum states into itself. We study the extension of an above quantum channel to the space of quantum states of general type that are convex combinations of normal states and singular states according to the Yosida–Hewitt decomposition. The interest to the study of quantum dynamics on the set of general quantum states arises in the consideration of a quantum dynamical semigroup acting in a Hilbert space of functions of infinite dimensional argument. In this case the above semigroup maps any pure vector quantum state into a state of general type. This effect can be considered in the example of averaging of quantum dynamical semigroup generated by a shift argument on a random Gaussian vector.

For the first time, a characterization is given of the circumstances under which an n-by-n matrix over a field has an (L)-(U) factorization. This is in terms of a comparison of ranks of the leading k-by-k principal submatrix to the rank of the first k columns and first k rows. Known results about special types of (L)-(U) factorizations follow as do some new results about near (L)-(U) factorization when a conventional (L)-(U) factorization does not exist. The proof allows explicit construction of an (L)-(U) factorization when one exists.

Recently, a substantial progress in studying the problem of optimal sampling recovery was made in a number of papers. In particular, this resulted in some progress in studying sampling recovery on function classes with mixed smoothness. Mostly, the case of recovery in the square norm was studied. In this paper we combine some of the new ideas developed recently in order to obtain progress in sampling recovery on classes with mixed smoothness in other integral norms.

In this work, we study Fourier multipliers on noncommutative spaces. In particular, we show a simple proof of (L^p)-(L^q) estimate of Fourier multipliers on general noncommutative spaces associated with semifinite von Neumann algebras. This includes the case of Fourier multipliers on general locally compact unimodular groups.

We define two types of approximate Roberts orthogonality with direction in the framework of a complex normed space. We examine their geometrical properties and demonstrate that the notion of (epsilon )-approximate directional orthogonality is weaker than that of (epsilon )-approximate orthogonality. Concerning the approximate Birkhoff orthogonality, we talk about the connection between them. Also, we provide the notion of an approximation Roberts directional orthogonality set and analyze the geometric characteristics of these sets. Furthermore, we discuss approximate orthogonality preserving mapping.

Several numerical radius inequalities in the framework of (C^*)-algebras are proved in this paper. These results, which are based on an extension of the Buzano inequality for elements in a pre-Hilbert (C^*)-module, generalize earlier numerical radius inequalities. An expression for the (C^*)-algebra-valued norm based on the numerical radius is also given.