Robust Bounds for Krylov Methods

In this paper we give bounds for polynomials of operators. These bounds are robust in low rank perturbations. This problem is encountered in the study of the convergence of Krylov methods. The central idea here is to view the resolvent as a meromorphic function.     

Operators on Soft Inner Product Spaces

In the present paper, self-adjoint operator and completely continuous operator on soft inner product spaces have been introduced and some basic properties of such operators are studied.     

General preservers of quasi-commutativity on self-adjoint operators

Let H be a separable Hilbert space and Bsa(H) the set of all bounded linear self-adjoint operators. We say that A,B∈Bsa(H) quasi-commute if there exists a nonzero ξ∈C such that AB=ξBA. Bijective maps on Bsa(H) which preserve quasi-commutativity in both directions are classified.     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

Symplectic Self-adjointness of Infinite Dimensional Hamiltonian Operators

Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown.     

Some properties of composition operators on Hilbert spaces of Dirichlet series

In this paper, we study some properties of composition operators on Hilbert spaces of Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen and MacCluer.     

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.