Search

The main focus of this paper is on the relationship between the spectrum of generators and the regularity of the fractional resolvent family. We will give a counter-example to show that the point-spectral mapping theorem is not valid for $ \{S_{\alpha}(t)\} $ if $ \alpha \neq 1 $; and we show that if $ \{S_{\alpha}(t)\} $ is stable, then we can determine the decay rate by $ \sigma(A) $ and some examples are given; we also prove that $ S_{\alpha}(t)x $ has a continuous derivative of order $ \alpha\beta > 0 $ if and only if $ x \in D(I-A)^{\beta} $. The main method we used here is the resolution of identity corresponding to a normal operator $ A $ and spectral measure integral.

In this paper, unitary symmetrizers are considered. It is well known that using Newton operatoralgorithm, similar to the usual Newton algorithm, for extracting the square root, one can provethat for every Hermitian operator T 0, there exists a unique Hermitian operator S 0 suchthat T = S2. Moreover, S commutes with every bounded operator R with which commutes T. Theoperator S is called a square root of the operator T and is denoted by T1=2. The existence of thesquare root allows one to determine the absolute value jTj = (TT)1=2 of the bounded operator T.For every bounded linear operator T : H ! H there exists a unique partially isometric operatorU : H ! H such that T = UjTj, KerU = KerT. Such an equality is called a polar expansionof the operator T. The Schmidt operator is understood as the unitary multiplier of the polarexpansion of a compact inverse operator, with the help of which E. Schmidt was the rst to obtainthe expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers.This paper shows that the unitary symmetrizer of an operator diers only in sign from the adjointSchmidt operator. The main result of the paper: if A is an invertible and compact operator, andS is a unitary operator such that the operator SA is self-adjoint, then the operator AS is alsoself-adjoint and the formula S = U holds, where U is the Schmidt operator.

In this paper, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. It is worth noticing that our first example says that the adjoint of a densely defined paranormal operator may have a trivial domain. In the end, we give an example of a densely defined closed one-to-one paranormal operator T whose adjoint is also injective and paranormal, but T is not normal.

Some stronger and equivalent metrics are defined on $\mathcal{M}$, the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $\mathcal{M}$ are investigated.

In this paper, we give a characterization of normality of Toeplitz operator Tφ on the Bergman space A2(D). First, we state basic properties for Toeplitz operator Tφ on A2(D). Next, we consider the normal Toeplitz operator Tφ on A2(D) in terms of harmonic symbols φ. Finally, we characterize the normal Toeplitz operators Tφ with non-harmonic symbols acting on A2(D).

We characterize normal Toeplitz operator on the Fock spaces F2(C). First, we state basic properties for Toeplitz operator Tφ on F2(C). Next, we study the normal Toeplitz operator Tφ on F2(C) in terms of harmonic symbols φ. Finally, we characterize the normal Toeplitz operators Tφ with non-harmonic symbols acting on F2(C).

We correct a mistake which affect our main results, namely the proof of Lemma 1. The main results of the article remain unchanged.

We investigate some numerical characteristics of Toeplitz operators including the numerical range, maximal numerical range and maximal Berezin set. Further, we establish an inequality for the Berezin number of an arbitrary operator on the Hardy–Hilbert space of the unit disc.

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

In this work, we describe all normal extensions of a multipoint minimal operators generated by linear multipoint differential-operator expressions for first order in the Hilbert space of vector functions, in terms of boundary values at the endpoints of infinitely many separated subintervals. Also we investigate compactness properties of the inverses of such extensions.

This paper develops an upper bound design method of the Lipschitz constant for the generalized Fermi–Dirac information entropy operator with a polyhedral admissible set. We introduce the concept of a normal operator from this class in which the constraint matrix has normalized columns. Next, we establish a connection between the normal and original operator. Finally, we demonstrate that the normal operator is majorized by the linear one and find numerical characteristics of this majorant.

We establish sufficient conditions for the solvability of a fourth order operator differential equation. These conditions are expressed in terms of operator coefficients.

A vector space X with algebra of all linear maps ∑(X) from X into itself and the ideal of all finite dimensional linear maps with dual (Conjugate) transformation T* to T from X' to itself form a relation in terms of relatively regular and linearly independent which is sufficient for mentioned title. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9342 BIBECHANA 10 (2014) 115-117

A vector space X with algebra of all linear maps ∑(X) from X into itself and the ideal of all finite dimensional linear maps with dual (Conjugate) transformation T* to T from X' to itself form a relation in terms of relatively regular and linearly independent which is sufficient for mentioned title. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9342 BIBECHANA 10 (2014) 115-117

The necessary and sufficient condition for equality to be attaind in the Kantrovich inequality is given and applied to an inequality of normal operators on Hilbert spaces.