Advances in Operator Theory最新文献

The maximal hyperplane section of the (l_infty ^n)-ball, i.e. of the n-cube, is the one perpendicular to (frac{1}{sqrt{2}} (1,1,0 ,ldots ,0)), as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the (l_p^n)-balls for very large (p ge 10^{15}). By Oleszkiewicz, Ball’s result does not transfer to (l_p^n) for (2< p < p_0 simeq 26.265). Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions n. Suppose that (p_0 le p < infty ). We show that the analogue of Ball’s result holds in (l_p^n)-balls for all hyperplanes with normal unit vectors a, if all coordinates of a have modulus (le frac{1}{sqrt{2}}) and p has distance (ge 2^{-p}) to the even integers. Under similar assumptions, we give a Gaussian upper bound for (20< p < p_0).

In this paper we present commutativity results for a general class of Szász–Mirakjan–Durrmeyer type operators and associated differential operators and investigate their eigenfunctions.Please confirm if the inserted city names are correct. Amend if necessary.The inserted city name is correct.

This paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Necessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained involving only the matrix entries.

We characterize the norm attainment set of a linear operator from ( ell _{infty }^{2}({mathbb {C}}) ) to ( ell _{1}^{2}({mathbb {C}}), ) with the help of a physical model involving two clocks entangled in a specific way. More generally, we introduce the (m, n)-clock Problem and establish its equivalence with computing the (ell _{infty }-ell _1) norm of an ( m times n ) matrix. We further give an explicit description of the smooth and the non-smooth points in ({mathbb {L}}big (ell _infty ^2({mathbb {C}}),ell _1^2({mathbb {C}})big ).)

In this study, singular value and norm inequalities for expressions of the form (SXT+Y) are established. It is shown that if (S,T,X,Y in mathcal {B(H)}) such that X, Y are compact operators, then

Additionally, we explore several applications of this inequality, which provide a broader framework for analysis and yield more nuanced insights. For (X, Yin mathcal {B(H)}) one notable application is the following inequality,

These results extend existing inequalities and offer new perspectives in operator theory.

Let X be a completely regular Hausdorff space and E and F be Banach spaces. Let (C_{rc}(X,E)) denote the Banach space of all continuous functions (f:Xrightarrow E) such that f(X) is a relatively compact set in E, and (beta _sigma ) be the strict topology on (C_{rc}(X,E)). We characterize dominated and absolutely summing operators (T:C_{rc}(X,E)rightarrow F) in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing ((beta _sigma ,Vert cdot Vert _F))-continuous operator (T:C_{rc}(X,E)rightarrow F) is dominated. Moreover, we obtain that every dominated operator (T:C_{rc}(X,E)rightarrow F) is absolutely summing if and only if every bounded linear operator (U:Erightarrow F) is absolutely summing.

We investigate how continuous linear functionals can be represented in terms of generic operators and certain kernels (Peano kernels), and we study lower bounds for the operators as a consequence, in the space of square-integrable functions. We apply and develop the theory for the Riemann–Liouville fractional derivative (an inverse of the Riemann–Liouville integral), where inequalities are derived with the Gaussian hypergeometric function. This work is inspired by the recent contributions by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024) and Jornet (Arch Math, 2024).

The program of matrix product states on tensor powers ({mathcal {A}}^{otimes {mathbb {Z}}}) of (C^*)-algebras is carried under the assumption that ({mathcal {A}}) is an arbitrary nuclear C*-algebra. For any shift invariant state (omega ), we demonstrate the existence of an order kernel ideal ({mathcal {K}}_omega ), whose quotient action reduces and factorizes the initial data (({mathcal {A}}^{otimes {mathbb {Z}}}, omega )) to the tuple (({mathcal {A}},{mathcal {B}}_omega = {mathcal {A}}^{otimes {mathbb {N}}^times }/{mathcal {K}}_omega , {mathbb {E}}_omega : text{AA }otimes {mathcal {B}}_omega rightarrow {mathcal {B}}_omega , {bar{omega }}: {mathcal {B}}_omega rightarrow {mathbb {C}})), where ({mathcal {B}}_omega ) is an operator system and ({mathbb {E}}_omega ) and ({bar{omega }}) are unital and completely positive maps. Reciprocally, given a (input) tuple (({mathcal {A}},{mathcal {S}},{mathbb {E}},phi )) that shares similar attributes, we supply an algorithm that produces a shift-invariant state on ({mathcal {A}}^{otimes {mathbb {Z}}}). We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite dimensional site algebras ({mathcal {A}}), such as the (C^*)-algebras of discrete amenable groups.

The envelope algorithm is used to precisely describe the numerical range of a block Toeplitz operator with 2-by-2 affine symbol in the case where the numerical range of the symbol at each point of the unit circle is a circular disk. In this setting, there is at most one flat portion on the boundary of the numerical range. Necessary and sufficient conditions are given for the flat portion to materialize.

We study the boundedness of multidimensional Hardy operators over (textbf{R}^n) in the framework of variable generalised local and global Morrey spaces with power-type weights, where we admit variable exponents for weights. We find conditions on the domain and target spaces ensuring such boundedness. In case of local spaces, these conditions involved values of variable integrability exponents of the domain and target spaces only at the origin and infinity. Due to the variability of the exponents of weights, the obtained results proved to be different corresponding to two distinct cases, which we called up to borderline and overbordeline case. We also pay special attention to a particular case, when the variable domain and target Morrey spaces are related to each other by Adams-type condition. The proofs are based on certain point-wise estimates for the Hardy operators, which allow, in particular, to get a statement on the boundedness from a local Morrey space to an arbitrary Banach function space with lattice property.