In this paper, we will provide a simple method for starting with a given finite frame for an $n$-dimensional Hilbert space $mathcal{H}_n$ with nonzero elements and producing a frame which is $epsilon$-nearly Parseval and $epsilon$-nearly unit norm. Also, the concept of the $epsilon$-nearly equal frame operators for two given frames is presented. Moreover, we characterize all bounded invertible ...

A geometric characterization is given for invertible quantum measurement maps. Denote by S(H) the convex set of all states (i.e., trace-1 positive operators) on Hilbert space H with dimH ≤ ∞, and [ρ1, ρ2] the line segment joining two elements ρ1, ρ2 in S(H). It is shown that a bijective map φ : S(H) → S(H) satisfies φ([ρ1, ρ2]) ⊆ [φ(ρ1), φ(ρ2)] for any ρ1, ρ2 ∈ S if and only if φ has one of the...

Let A be a commutative comodule algebra over a commutative bialgebra H . The group of invertible relative Hopf modules maps to the Picard group of A, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring A⊗H , or as a Harrison cohomology group. Our methods are based on elementary K-theory. The Hilbert 90 Theorem follows as a corollary. The p...

We investigate algebras of block-symmetric analytic functions on spaces $\ell_{p}(\mathbb{C}^s)$ which are $\ell_{p}$-sums $\mathbb{C}^{s}.$ consider properties algebraic bases polynomials,intertwining operations spectra the and representations as a semigroup exponential type several variables. All invertible elements described for case $p=1.$

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H , and we analyze the set of attainable frame bounds. In the case whereH is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite dimensional case is a result we have proven which concerns the decomposition of a posi...

Consider an invertible n × n matrix over some field. The Gauss-Jordan elimination reduces this matrix to the identity matrix using at most n row operations and in general that many operations might be needed. In [1] the authors considered matrices in GL(n, q), the set of n × n invertible matrices in the finite field of q elements, and provided an algorithm using only row operations which perfor...

In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

If a ring element α = 1 − ab is Drazin invertible, so is β = 1 − ba. We show that the Drazin inverse of β is expressible by a simple formula (in generalization of “Jacobson’s Lemma”) in terms of the Drazin inverse and the spectral idempotent of α. The commutation rules αa = a β and b α = β b are extended to the Drazin inverses, and the spectral idempotents of α and β are shown to be isomorphic....

We briefly survey results about Invertible Algebras (algebras having bases that consist entirely of units) and other related notions. In addition, we consider the existence of an augmentation map as a possible way in which results about group rings, the archetypical invertible algebras, may be extended to more general settings. 1 Invertible Algebras and Some Related Notions Section 1 of this pa...