To a generalized tight continuous frame in a Hilbert space H, indexed by a locally COMPACT space S endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on S in spaces of vectors comparable with H. If the continuous frame is provided by the action of a suitable family of bounded OPERATORs on a fixed window, a symbolic calculus emerges, assigning OPERATORs in H to functions on S. We give some criteria of relative COMPACTness for sets and for families of COMPACT OPERATORs, involving tightness properties in terms of objects canonically associated to the frame. Particular attention is dedicated to a magnetic version of the pseudo differential calculus.

In this paper we derive some improvements of means inequalities for Hilbert space OPERATORs. More precisely, we obtain refinements and reverses of the arithmetic-geometric OPERATOR mean inequality. As an application, we also deduce an improved variant for the refined arithmetic-Heinz mean in- equality. We also present some eigenvalue inequalities for differences of certain OPERATOR means.

In this paper, we study weakly COMPACT left multipliers on the Banach algebra LUC (G)*. We show that G is COMPACT if and only if there exists a non-zero weakly COMPACT left multipliers on LUC (G)*. We also investigate the relation between positive left weakly completely continuous elements of the Banach algebras LUC (G)* and L¥ (G)*. Finally, we prove that G is finite if and only if there exists a non-zero multiplicative linear functional m on LUC (G) such that m is a left weakly completely continuous elements of LUC (G)*.

We shall provide a brief account on new achievements in the study of composition and composition-differentiation OPERATORs acting on classical spaces of analytic functions on the unit disk, as well as on the polydisk and the unit ball. The emphasis is given to the Hardy space, the Bergman space, and the Dirichlet space on the unit disk in complex plane. In the last section, we shall provide some new results regarding the Hilbert-Schmidt OPERATORs in the setting of several complex variables.

An OPERATOR T on a complex Hilbert space H is said to be skew symmetric if there exists a conjugate-linear, isometric involution C:H®H such that CTC=-T*. In this paper, using an interpolation theorem related to conjugations, we give a geometric characterization for a class of OPERATORs to be skew symmetric. As an application, we get a description of skew symmetric partial isometries.