On (n,m)(n,m)-AA-normal and (n,m)(n,m)-AA-quasinormal semi-Hilbertian space operators

summary:The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces, i.e. spaces generated by positive semi-definite sesquilinear forms. Let ${\mathcal H}$ be a Hilbert space and let $A$ be a positive bounded operator on ${\mathcal H}$. The semi-inner product $\langle h\mid k\rangle _A:=\langle Ah\mid k\rangle$, $h,k \in {\mathcal H}$, induces a semi-norm $\|{\cdot }\|_A$. This makes ${\mathcal H}$ into a semi-Hilbertian space. An operator $T\in {\mathcal B}_A({\mathcal H})$ is said to be $(n,m)$-$A$-normal if $[T^n,(T^{\sharp _A})^m]:=T^n(T^{\sharp _A})^m-(T^{\sharp _A})^mT^n=0$ for some positive integers $n$ and $m$

(m,p)(m, p)-hyperexpansive mappings on metric spaces

In the present paper, we define the concept of $(m,p)$-hyperexpansive mappings in metric space,which are the extension of $(m,p)$-isometric mappings recently introduced in $[13].$ We give a first approach of the general theory of these maps

Spectral properties of (m;n)-isosymmetric multivariable operators

Inspired by recent works on $m$-isometric and $n$-symmetric multivariables operators on Hilbert spaces, in this paper we introduce the class of $(m, n)$-isosymmetric multivariables operators. This new class of operators emerges as a generalization of the $m$-isometric and $n$-isosymmetric multioperators. We study this class of operators and give some of their basic properties. In particular, we show that if ${\bf \large R} \in {\mathcal B}^{(d)}({\mathcal H})$ is an $(m,n )$-isosymmetric multioperators and ${\bf \large Q}\in {\mathcal B}^{(d)}({\mathcal H})$ is an $q$-nilpotent multioperators, then ${\bf\large R} +{\bf\large Q}$ is an $(m + 2q - 2,n+2q-1)$-isosymmetric multioperators under suitable conditions. Moreover, we give some results about the joint approximate spectrum of an $(m,n)$-isosymmetric multioperators

Optimal control problems governed by a class of nonlinear systems

This article suggested a solution to a flow control problem governed by a class of nonlinear systems called bilinear systems. The problem was initially well-posed, and after it was established that an optimal control solution existed, its characteristics were stated. After that, we demonstrated how to use various bounded feedback controls to make a plate equation's flow close to the required profile. As an application, we resolved the plate equation-governed partial flow control issue. The findings bring up a variety of system applications, which can be employed in engineering advancement

On Some Normality-Like Properties and Bishop's Property () for a Class of Operators on Hilbert Spaces

We prove some further properties of the operator ∈[QN] (-power quasinormal, defined in Sid Ahmed, 2011). In particular we show that the operator ∈[QN] satisfying the translation invariant property is normal and that the operator ∈[QN] is not supercyclic provided that it is not invertible. Also, we study some cases in which an operator ∈[2QN] is subscalar of order ; that is, it is similar to the restriction of a scalar operator of order to an invariant subspace

A-m-Isometric operators in semi-Hilbertian spaces

AbstractIn this work, the concept of m-isometry on a Hilbert space are generalized when an additional semi-inner product is considered. This new concept is described by means of oblique projections