Advances in Operator Theory最新文献

A substantial theory of the Karcher mean exists in the settings of Riemannian manifolds and positive matrix and operator spaces. Here a general setting for the study of the Karcher mean on Lie groups is proposed. Local existence and uniqueness results already exist, but here, a significant global result is obtained. It is shown that a natural computable iteration scheme for the Karcher mean exists for the Lie group of upper triangular unipotent matrices and that it always converges starting from any point after finitely many steps.

The strong law of large numbers for a double sequence of pairwise M-dependent random variables is established. An extension to Kuratowski-convergence of the strong law of large numbers for a double sequence of pairwise M-dependent multivalued random variables with closed values is stated.

In this paper, we study the regularizing effects of a singular first-order term in some degenerate elliptic equations with zero-order term involving Hardy potential. The model problem is

where (Omega ) is a bounded open subset in ({mathbb {R}}^{N}) with (0in Omega ), (gamma ge 0), (1<p<N), (0<theta <1), and (0<r<p-theta ). We prove existence and regularity results for solutions under various hypotheses on the datum f.

In this article, we introduce a geometrical notion, called property strongly UC which is stronger than property UC and prove the existence of best approximations for a new class of almost cyclic (psi)-contraction maps defined on a pair of subsets of a metric space. As a particular case of this existence theorem, we obtain the main results of [Sadiq Basha, S., Best approximation theorems for almost cyclic contractions. J. Fixed Point Theory Appl. 23 (2021)] and [Eldred, A. Anthony; Veeramani, P., Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2006)]. Moreover, we study the existence of a best approximation and continuity properties of almost cyclic contractions in the context of a reflexive Banach space and a metric space.

In this paper, we prove interpolating numerical radius inequalities involving the generalized numerical radii induced by unitarily invariant norms. These inequalities refine well-known interpolating norm inequalities. Several related numerical radius inequalities are also established.

We introduce a construction of Dirichlet forms on von Neumann algebras M associated to any eigenvalue of the Araki modular Hamiltonian of a faithful normal non-tracial state, providing also conditions by which the associated Markovian semigroups are GNS symmetric. The structure of these Dirichlet forms is described in terms of spatial derivations. Coercivity bounds are proved and the spectral growth is derived. We introduce a regularizing property of positivity preserving semigroups (superboundedness) stronger than hypercontractivity, in terms of the symmetric embedding of M into its standard space (L^2(M)) and the associated noncommutative (L^p(M)) spaces. We prove superboundedness for a special class of positivity preserving semigroups and that some of them are dominated by the Markovian semigroups associated to the Dirichlet forms introduced above, for type I factors M. These tools are applied to a general construction of the quantum Ornstein–Uhlembeck semigroups of the Canonical Commutation Relations CCR and some of their non-perturbative deformations.

In this paper, we prove some new singular value and unitarily invariant norm inequalities for matrices. Among other results, it is shown that if X, Y, Z, W are n (times ) n matrices, then

and

for (j=1,2,ldots ,n), where (left| cdot right| ,w(cdot ),) and ( s_{j}(cdot )) denote the spectral norm, the numerical radius, and the jth singular value of matrices.

Let ( {mathcal {A}} ) be a unital (*)-algebra with characteristic not 2 and containing a nontrivial projection. We show that each nonlinear (*)-Lie derivation on ({mathcal {A}}) is a linear (*)-derivation. Moreover, we characterize nonlinear left (*)-Lie centralizers and nonlinear generalized (*)-Lie derivations. These results are applied to standard operator algebras and von Neumann algebras in complex Hilbert spaces, which generalize some known results.

Generalizing the notions of the row and the column stochastic matrices, we introduce the multidimensional Q-stochastic tensors. We prove that every Q-stochastic tensor can be decomposed as a convex combination of finitely many binary Q-stochastic tensors and that the binary Q-stochastic tensors are exactly the extreme points of the compact convex set of all Q-stochastic tensors with the same size. Applications to characterizing the Bell locality of a quantum state in a multipartite system are demonstrated. Algorithms for computing the convex decompositions of Q-stochastic tensors are provided.

Let ( T in mathcal {B}(mathcal {H})) be a bounded linear operator on a Hilbert space ( mathcal {H}), and let ( T = U vert T vert ) be the polar decomposition of T. For any (r > 0), the transform (S_{r}(T)) is defined by (S_{r}(T) = U vert T vert ^{r} U). In this paper, we discuss the transform (S_{r}(T)) of some classes of operators such as p-hyponormal and rank one operators. We provide a new characterization of invertible normal operators via this transform. Afterwards, we investigate when an operator T and its transform ( S_{r}(T) ) both have closed ranges, and show that this transform preserves the class of EP operators. Finally, we present some relationships between an EP operator T, its transform ( S_{r}(T)) and the Moore–Penrose inverse ( T^{+} ).