Let T ? B(H) be a bounded linear operator on Hilbert space H, and let = U|T| its polar decomposition. Then, for every [0,1] the ?-Aluthge transform of is defined by ??(T) |T|?U|T|1-?. In this paper, we characterize invertible, binormal, EP operators intersection with special class introduced via transform.

This paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.

Let m,n, p be positive integers such that m ≥ n + p. Suppose (A,B) ∈ Cm×n ×Cm×p, and let P(A,B) = {(E,F ) ∈ Cm×n ×Cm×p : there is X ∈ Cn×p such that (A− E)X = B − F}. The total least square problem concerns the determination of the existence of (E,F ) in P(A,B) having the smallest Frobenius norm. In this paper, we characterize elements of the set P(A,B) and derive a formula for ρ(A,B) = inf {‖[...

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L = A +V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles betw...

An interesting result proved by Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968) is that the set of irreducible operators dense $${\mathcal {B}}({\mathcal {H}})$$ sense Hilbert-Schmidt approximation. In a von Neumann algebra {M}}$$ with separable predual, an operator $$a\in {\mathcal said to be if $$W^*(a)$$ subfactor , i.e., $$W^*(a)'\cap {M}}={{\mathbb {C}}} \cdot I$$ . Let $$...

Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then, the Aluthge transform is defined by ∆ (T ) = |T |U |T |. Let ∆n(T ) denote the n-times iterated Aluthge transform of T , i.e. ∆0(T ) = T and ∆n(T ) = ∆(∆n−1(T )), n ∈ N. We prove that the sequence {∆n(T )}n∈N converges for every r× r diagonalizable matrix T . We show that the limit ∆∞(·) is a map of class C∞...

This paper concerns spectral stability and time evolution of N -solitons in the KdV hierarchy with mixed commuting time flows. Spectral stability problem is analyzed by using a pair of self-adjoint operators with finite numbers of negative eigenvalues. We show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators i...

Abstract Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper considers discrete Tracy–Widom operators, and gives sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kerne...

The article is devoted to building various dilatations of linear operators. explicit construction a unitary dilation compression operator considered. Then the J-unitary dilatation bounded constructed by means knot concept operator. Using Pavlov method, we construct self-adjoint dissipative We consider spectral and translational representations densely defined with nonempty set regular points. a...