We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 HV for some self-adjoint operator H; (ii) the operators H and H are resolvent ...

In the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. In particular, it isshown that the Cauchy-Schwarz inequality holds. Moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy Hilbert space has a complementary subspace.Finally, the notions of fuzzy bo...

In this article we study two different generalizations of von Neumann regularity, namely strong topological regularity and weak regularity, in the Banach algebra context. We show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. Then we consider strong topological regularity of certain concrete algebras. Moreover we obtain ...

The notion of $k$-frames was recently introduced by Gu avruc ta in Hilbert  spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous super positions. In this manuscript, we construct a continuous $k$-frame, so called c$k$-frame along with an atomic system ...

Given a conditionally completely positive map L on a unital ∗-algebra A, we find an interesting connection between the second Hochschild cohomology of A with coefficients in the bimodule EL = Ba(A⊕M) of adjointable maps, where M is the GNS bimodule of L, and the possibility of constructing a quantum random walk (in the sense of [2, 11, 13, 16]) corresponding to L.

where A is the infinitesimal generator of the C0-semigroup T (t) on the state space X, B is a bounded linear operator from input space U to X, C is a bounded linear operator from X to the output space Y , and D is a bounded operator from U to Y . The spaces X, U and Y are assumed to be Banach spaces. More detail on the system (1) can be found in Curtain and Zwart [1]. For the system (1) we intr...

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator T on the Segal-Bargmann space, the Berezin transform of T is a function whose partial derivatives of all orders are bounded. Similarly, if T is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined “invariant derivatives”...

amalgam space W (C,L2(R, T 1 1 )) which is locally continuous and globally L2. We show that the sampling or discretization operator R from S to l2(Z, T 1 1 ) is a bounded linear operator. We introduce the dilated spaces S∆ = D∆ S parametrized by the coarseness ∆, and show that the discretization operator is also bounded with a bounded inverse for any ∆ ∈ Zn. This allows us to represent discrete...

Jordan C*-algebras go back to Kaplansky, see [20]. Let J be a complex Banach Jordan algebra, that is, a complex Banach space with commutative bilinear product x◦y satisfying x◦(x2◦y) = x2◦(x◦y) as well as ||x◦y|| ≤ ||x||·||y||, Bounded symmetric domains and generalized operator algebras 51 and suppose that on J is given a (conjugate linear) isometric algebra involution x 7→ x∗. Then J is called...