Hyponormal Operators Research Articles

A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators

A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators

New extension of quasi-$ M $-hypnormal operators

This study introduces a new class of operators called polynomilally quasi-$ M $-hyponormal, which combining $ M $-hyponormal, quasi-$ M $-hyponormal, and $ k $-quasi-$ M $-hyponormal operators. We will demonstrate several properties of this class that correspond to those of $ k $-quasi-$ M $-hyponormal operators.

Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results

<p>In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.</p>

On the M�bius invariant principal functions of Pincus

In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator \(T\) with \(\operatorname{rank} (T^*T-TT^*) =1\), modulo unitary equivalence, is the unilateral shift.

New Generalizations for Ϻ-Hyponormal Operators

This article contains a new generalizations of Ϻ-hyponormal operators which is namely (Ϻ,θ)-hyponormal operator define on Hilbert space H. Furthermore, we investigate some properties of this concept such as the product and sum of two (Ϻ, θ)-hyponormal operators, At the end the operator equation where , has been used for getting several characterization of (Ϻ,θ)-hyponormal operators.

Spherical Aluthge transform, spherical p and log-hyponormality of commuting pairs of operators II

In this paper, we consider the notions of spherical p, log, and w-hyponormality of commuting pairs of operators and study their properties and relations. We first show that spherically p-hyponormal 2-variable weighted shifts of hyponormal operators are invariant under powers and the generalized spherical Aluthge transformation. Second, we study the relation between spherical p-hyponormality and spherical w-hyponormality. Third, we study whether the spherical p-hyponormality with 0<p<1 is invariant under the spherical Aluthge transformation. Finally, we briefly discuss the relation between log-hyponormality in single operators and spherically log-hyponormality in commuting pairs of operators.

Unbounded operators having self‐adjoint, subnormal, or hyponormal powers

AbstractWe show that if a densely defined closable operator A is such that the resolvent set of A2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that is normal for some , then T is normal. Hence a closed subnormal operator T such that is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that and are self‐adjoint for some coprime numbers p and q, then A must be self‐adjoint.

N-hyponormality and n-contractivity of generalized Bergman weighted shifts

Several bridges of operator properties between subnormal and hyponormal operators on Hilbert spaces have been developed well. In the class of contractive operators, we may consider two bridges consisting of n-hyponormal and n-contractive operators for n∈N, where N is the set of positive integers. Several operator theorists have studied the one-step backward extension of generalized Bergman weighted shifts such as Agler shifts, Bergman-like shifts, and Bergman-type weighted shifts to detect gap for these two bridges. Obviously the class of homographic shifts S(a,b,c,d) contains these all generalized Bergman weighted shifts above. In this paper we obtain some equivalent conditions for the n-hyponormality and n-contractivity of the one-step backward extension of S(a,b,c,d), which may induce the known results about Agler shifts, Bergman-like shifts, and Catalan-like weighted shifts etc.

The Bishop’s property (β) for class A operators

We say that an operator T on a Hilbert space H has the Bishop?s property (?) if for an arbitrary open set U ? C and analytic functions fn : U ? H with ?(T ? z) fn(z)? converges to 0 uniformly on every compact subset of U as n ? ?then ? fn? converges to 0 uniformly on every compact subset ofU as n ? ?. An operator T on H is called to be hyponormal if T*T ? TT*, and T is called to be class A if T*T ? |T2|. In this papaer, we give an elementary proof of the assertion that every hyponormal operator has the Bishop?s property (?). And we show that every class A operator has the Bishop?s property (?). Moreover, we also show a class A operator T is similar to a hyponormal operator if T is invertible, and hence T has the growth condition (G1).

Matrix-valued Gabor frames over LCA groups for operators

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

Posinormal composition operators on H2

A bounded linear operator A on a Hilbert space is posinormal if there exists a positive operator P such that AA⁎=A⁎PA. Posinormality of A is equivalent to the inclusion of the range of A in the range of its adjoint A⁎. Every hyponormal operator is posinormal, as is every invertible operator. We characterize both the posinormal and coposinormal composition operators Cφ on the Hardy space H2 of the open unit disk D when φ is a linear-fractional selfmap of D. Our work reveals that there are composition operators that are both posinormal and coposinormal yet have powers that fail to be posinormal.

Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems.

KARAKTERISTIK OPERATOR PARANORMAL- * QUASI

Given Hilbert space H over the fields of . This study aimed to investigate the paranormal- * quasi operators and their properties in Hilbert space. The study resulted the properties of paranormal- * quasi operators, hyponormal operator, class A operator, Class A- * operator, p- hyponormal operator for p &gt; 0, - paranormal operators, compact operator, and the relationship between them

SOME PROPERTIES OF (n, m)-POWER-HYPONORMAL IN SEMI HILBERTIAN SPACE

In [1] (Kikete Wabuya, Luketero Wanyonyi, and Justus Mile) show that if an operator (n,m) hyponormal is isometrically equivalent to an operator S, then S is also (n ,m) hyponormal operator. In this paper, we prove results in the same spirit but in a semi Hilbertian space, i.e., spaces generated by positive semi-definite sesquilinear forms. This kind of spaces appears in many problems concerning linear and bounded operators on Hilbert spaces and is intensively studied in the present; some of the basic properties of this class are studied. Moreover, the product, tensor product and the sum of finite numbers of this type are discussed.

VERSIONS OF FUGLEDE-PUTNAM THEOREM ON p-w-HYPONORMAL OPERATORS

The aim of the article is the presentation of certain extensions of the famous Fuglede-Putnam Theorem on the class of $p$-$w$-hyponormal operators, which generalize some results proved by authors in \cite{Prasad}.

Fuglede – Putnam type theorems for extension of -hyponormal operators

UDC 517.9 We consider -quasi--hyponormal operator suchthat for some and prove the Fuglede–Putnam type theorem when adjoint of is -quasi--hyponormal or dominant operators.We also show that two quasisimilar -quasi--hyponormal operators have equal essential spectra.

(h, M)-Hyponormal Operators

In this work, we introduce a study on a new generalization for hyponormal op-erators, which is (h, M)- hyponormal operator. Also, we given some properties for this concept, as well as, we solve the - commuting operator equation for (h, M)- hyponormal operators

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Numerical range of quaternionic right linear bounded operators

In this paper, we prove that for a right linear bounded operator on a quaternionic Hilbert space, the norm and the numerical radius are equal if and only if the norm and the spectral radius are equal. We also show that the spherical spectrum of a quaternionic bounded operator is included in the closure of its numerical range, and we show that the numerical range of an operator on a quaternionic Hilbert space is not necessarily convex. For a quaternionic bounded normal operator, we prove that the convex hull of the closure of its numerical range is equal to the convex hull of its spherical spectrum. Finally, we give some inequalities between the numerical radius, the spectral radius and the norm of a right linear bounded operator, and we prove also that the norm and the numerical radius of a quaternionic bounded hyponormal operator are equal.

Hyponormal operators in soft sets

In this paper, we introduce the notion of hyponormal operators in soft Hilbert spaces, some properties of these operators are studied, and in a similar way to [6], a hyponormal soft operator is built from a family of operators. In addition, some results are obtained for self-adjoint, invertible, unitary, unit equivalent and normal soft linear operators that relate the properties of these soft operators with a family of operators in certain Hilbert spaces.