Advances in Operator Theory最新文献

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.

In this paper it is studied the well-posedness in several senses for non-homogeneous Cauchy problems where the infinitesimal generator depends on the time parameter. More specifically, we analyze the existence of classical, mild and weak solutions and their relationships. Thanks to uniqueness arguments, mild solutions are proved to satisfy a classical variational formulation. Finally, these results are applied to a thermoelastic plate model where the thermal part is of Cattaneo type and all the physical coefficients depend on time.

We showed two types of operator inequalities between the ((n+1))th residual relative operator entropy and the difference of the nth residual relative operator entropies. They are similar partially but have some differences. We investigate what these differences come from. Inequalities other than the previous ones are given through this process.

First, we present a method for obtaining a canonical set of root functions and Jordan chains of the invertible matrix polynomial L(z) through elementary transformations of the matrix L(z) alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations (Lleft( frac{d}{dt}right) u=0), where u(t) is n-dimensional unknown function. We illustrate the effectiveness of this method by applying it to solve a high-order linear system of ODEs. Second, given a matrix generalized Nevanlinna function (Qin N_{kappa }^{n times n}), that satisfies certain conditions at (infty ), and a canonical set of root functions of (hat{Q}(z):= -Q(z)^{-1}), we construct the corresponding Pontryagin space ((mathcal {K}, [.,.])), a self-adjoint operator (A:mathcal {K}rightarrow mathcal {K}), and an operator (Gamma : mathbb {C}^{n}rightarrow mathcal {K}), that represent the function Q(z) in a Krein–Langer type representation. We illustrate the application of main results with examples involving concrete matrix polynomials L(z) and their inverses, defined as (Q(z):=hat{L}(z):= -L(z)^{-1}).

Let (mathcal {T}) denote the algebra of all (2 times 2) upper triangular matrices over a field (mathbb {F}). We show that the linear space of all 2-local derivations on (mathcal {T}) decomposes as (mathcal {L} = mathcal {D} oplus mathcal {L}_0), where (mathcal {D}) is the subspace of all derivations, and (mathcal {L}_0) consists of 2-local derivations vanishing on a subset of (mathcal {T}), isomorphic to the space of functions (f:mathbb {F}rightarrow mathbb {F}) such that (f(0)=0). For any 2-local automorphism (Lambda ) on (mathcal {T}), we show that there exists a unique automorphism (phi ) and a 2-local automorphism (Lambda _{1} in varPsi ) such that (Lambda = phi Lambda _1), where (varPsi ) is the monoid of 2-local automorphisms that act as the identity on a subset of (mathcal {T}). Furthermore, we establish that (varPsi ) is isomorphic to the monoid of injective functions from (mathbb {F}^{*}) to itself.

This paper is devoted to the study of two classes of operators related to disjointly weakly compact sets, which we call DW-DP operators and DW-limited operators, respectively. They carry disjointly weakly compact sets in a Banach lattice onto Dunford–Pettis sets and limited sets, respectively. We show that DW-DP (resp. DW-limited) operators are precisely those operators which are both weak Dunford–Pettis and order Dunford–Pettis (resp. weak(^*) Dunford–Pettis and order limited) operators. Furthermore, the approximation properties of positive DW-DP and positive DW-limited operators are given.

Often, when we consider the time evolution of a system, we resort to approximation: Instead of calculating the exact orbit, we divide the time interval in question into uniform segments. Chernoff’s results in this direction provide us with a general approximation scheme. There are situations when we need to break the interval into uneven pieces. In this paper, we explore alternative conditions to the one found by Smolyanov et al. such that Chernoff’s original result can be extended to unevenly distributed time intervals. Two applications concerning the foundations of quantum mechanics and the central limit theorem are presented.