Advances in Operator Theory最新文献

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.

This paper considers paired operators in the context of the Lebesgue Hilbert space (L^2) on the unit circle and its subspace, the Hardy space (H^2.) The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.

Tingley’s problem asks whether every surjective isometry between two unit spheres of Banach spaces can be extended to a surjective real linear isometry between the whole spaces. Let ({A_mu }_{mu in M}) and ({A_{nu }}_{nu in N}) be two collections of uniformly closed extremely C-regular subspaces. In this paper, we prove that if (Delta ) is a surjective isometry between two unit spheres of (ell ^1)-sums of uniformly closed extremely C-regular subspaces ({A_{mu }}_{mu in M}) and ({A_{nu }}_{nu in N}), then (Delta ) admits an extension to a surjective real linear isometry between the whole spaces. Typical examples of such Banach spaces B are (C^1(I)) of all continuously differentiable complex-valued functions on the closed unit interval I equipped with the norm (Vert fVert _{1}=|f(0)|+Vert f'Vert _{infty }) for (fin C^1(I)), (C^{(n)}(I)) of all n-times continuously differentiable complex-valued functions on I with the norm (Vert fVert _{1}=sum _{k=0}^{n-1}|f^{(k)}(0)|+~Vert f^{(n)}Vert _{infty }) for (C^{n}(I)), and (ell ^1(mathbb {N})) of all complex-valued functions on the set (mathbb {N}) of all natural numbers with the norm (Vert aVert _{1}=sum _{nin mathbb {N}}|a(n)|) for (ain ell ^1(mathbb {N})).

The Collatz map (or the (3n{+}1)-map) f is defined on positive integers by setting f(n) equal to (3n+1) when n is odd and n/2 when n is even. The Collatz conjecture states that starting from any positive integer n, some iterate of f takes value 1. In this study, we discuss formulations of the Collatz conjecture by (C^{*})-algebras in the following three ways: (1) single operator, (2) two operators, and (3) Cuntz algebra. For the (C^{*})-algebra generated by each of these, we consider the condition that it has no non-trivial reducing subspaces. For (1), we prove that the condition implies the Collatz conjecture. In the cases (2) and (3), we prove that the condition is equivalent to the Collatz conjecture. For similar maps, we introduce equivalence relations by them and generalize connections between the Collatz conjecture and irreducibility of associated (C^{*})-algebras.

The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of the operator, and we propose a partial ordering for the class of all bounded linear operators which lead to ill-posed operator equations. For compact linear operators, there is a simple characterization in terms of the decay rates of the singular values. In the context of the validity of the spectral theorem the partial ordering can also be understood. We highlight that range inclusions yield partial ordering, and we discuss cases when compositions of compact and non-compact operators occur. Several examples complement the theoretical results.

Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire category theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple ((A_0, A_1).) Lions problem, posed in 1960s, is to prove that different parameters ((theta ,p)) produce different interpolation spaces ((A_0, A_1)_{theta , p}.) We first establish conditions on (A_0) and (A_1) so that interpolation spaces of this couple are strictly intermediate spaces between (A_0+A_1) and (A_0cap A_1.) This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if (pne p^*) the operator ideals ({mathcal {L}}^{(a)}_{p,q}(X,Y),) ({mathcal {L}}^{(a)}_{p^*,q^*}(X,Y)) generated by approximation numbers are distinct. Moreover, for any fixed p,  either all operator ideals ({mathcal {L}}^{(a)}_{p,q}(X,Y)) collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch’s problem for operator ideals based on general s-numbers. However, the BCT can be used to prove a lethargy result for arbitrary s-numbers which guarantees that, under very minimal conditions on X, Y,  the space ({mathcal {L}}^{(s)}_{p,q}(X,Y)) is strictly embedded into ({mathcal {L}}^{mathcal {A}}(X,Y).)