We present a general approach to a module frame theory in C∗algebras and Hilbert C∗-modules. The investigations rely on the ideas of geometric dilation to standard Hilbert C∗-modules over unital C∗-algebras that possess orthonormal Hilbert bases, of reconstruction of the frames by projections and by other bounded module operators with suitable ranges. We obtain frame representation and decompos...

A notion of curvature is introduced in multivariable operator theory. The curvature invariant of a Hilbert module over C[z(1),., z(d)] is a nonnegative real number which has significant extremal properties, which tends to be an integer, and which is hard to compute directly. It is shown that for graded Hilbert modules, the curvature agrees with the Euler characteristic of a certain finitely gen...

A notion of curvature is introduced in multivariable operator theory, that is, for commuting d tuples of operators acting on a common Hilbert space whose “rank” is finite in an appropriate sense. The curvature invariant is a real number in the interval [0, r] where r is the rank, and for good reason it is desireable to know its value. For example, there are significant and concrete consequences...

in this paper, we state some results on product of operators with closed rangesand we solve the operator equation txs*- sx*t*= a in the general setting of theadjointable operators between hilbert c*-modules, when ts = 1. furthermore, by usingsome block operator matrix techniques, we nd explicit solution of the operator equationtxs*- sx*t*= a.

Let H m be the reproducing kernel Hilbert space with the kernel function (z, w) ∈ B×B → (1− m ∑ i=1 ziw̄i) . We show that if θ : B → L(E , E∗) is a multiplier for which the corresponding multiplication operator Mθ ∈ L(H m ⊗ E , H 2 m ⊗ E∗) has closed range, then the quotient module Hθ, given by · · · −→ H m ⊗ E Mθ −→ H m ⊗ E∗ πθ −→ Hθ −→ 0, is similar to H m ⊗F for some Hilbert space F if and on...

‎Frames generalize orthonormal bases and allow representation of all the elements of the space‎. ‎Frames play significant role in signal and image processing‎, ‎which leads to many applications in informatics‎, ‎engineering‎, ‎medicine‎, ‎and probability‎. ‎In this paper‎, ‎we introduce the concepts of operator frame for the space $End_{mathcal{A}}^{ast}(mathcal{H})$ of all adjointable operator...

In this paper, we introduce the concept of $g$-dual frames for Hilbert $C^{*}$-modules, and then the properties and stability results of $g$-dual frames  are given.  A characterization of $g$-dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of $g$-dual frames for Riesz bases in Hilbert spaces is ...

Let A d be the complex polynomial ring in d variables. A contractive A d-module is Hilbert space H equipped with an A d action such that for any ξ 1 , ξ 2 ,. Such objects have been shown to be useful for modeling d-tuples of mutually commuting operators acting on a Hilbert space. An example is given by H 2 d , which is the Hilbert space of holomorphic functions on B d = {z ∈ C d : |z| ≤ 1} gene...

In this paper we consider C0-group of unitary operators on a Hilbert C*-module E. In particular we show that if A?L(E) be a C*-algebra including K(E) and ?t a C0-group of *-automorphisms on A, such that there is x?E with =1 and ?t (?x,x) = ?x,x t?R, then there is a C0-group ut of unitaries in L(E) such that ?t(a) = ut a ut*.

In this paper, we characterize rank one preserving module maps on a Hilbert C∗−module and study its applications on free probability theory.