scholarly journals On Subscalarity of Some 2 × 2M-Hyponormal Operator Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Fei Zuo ◽  
Junli Shen
Keyword(s):  
Type Theorem ◽  
Hyponormal Operator ◽  
Operator Matrices ◽  
Hyponormal Operators

We provide some conditions for2×2operator matrices whose diagonal entries areM-hyponormal operators to be subscalar. As a consequence, we obtain that Weyl type theorem holds for such operator matrices.

scholarly journals HYPONORMAL OPERATORS, WEIGHTED SHIFTS AND WEAK FORMS OF SUPERCYCLICITY

2006 ◽  
Vol 49 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Frédéric Bayart ◽  
Etienne Matheron
Keyword(s):  
Banach Space ◽  
Unitary Operator ◽  
Dimensional Subspace ◽  
The Other ◽  
Hyponormal Operator ◽  
Weighted Shifts ◽  
One Dimensional ◽  
Other Hand ◽  
Hyponormal Operators

AbstractAn operator $T$ on a Banach space $X$ is said to be weakly supercyclic (respectively $N$-supercyclic) if there exists a one-dimensional (respectively $N$-dimensional) subspace of $X$ whose orbit under $T$ is weakly dense (respectively norm dense) in $X$. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never $N$-supercyclic. Finally, we characterize $N$-supercyclic weighted shifts.

scholarly journals On the spectrum of n-tuples of p-hyponormal operators

1998 ◽  
Vol 40 (1) ◽  
pp. 123-131 ◽  
Author(s):  
B. P. Duggal
Keyword(s):  
Spectral Properties ◽  
Polar Decomposition ◽  
Partial Isometry ◽  
Hyponormal Operator ◽  
Square Root ◽  
Linear Transformations ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Algebra Of Operators ◽  
Number Of Authors

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.

scholarly journals Spectral properties of square hyponormal operators

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4845-4854
Author(s):  
Muneo Chō ◽  
Dijana Mosic ◽  
Biljana Nacevska-Nastovska ◽  
Taiga Saito
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Spectral Properties ◽  
Complex Hilbert Space ◽  
Hyponormal Operator ◽  
Bounded Linear ◽  
Basic Properties ◽  
Hyponormal Operators ◽  
Eigen Values

In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis ?(T) ? (-?(T)) ? {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x,y ? H are corresponding eigen-vectors, respectively, then ?x,y? = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators.

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

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Salah Mecheri
Keyword(s):  
Invariant Subspace ◽  
Operator Theory ◽  
Affirmative Answer ◽  
Hyponormal Operator ◽  
Hyperinvariant Subspace ◽  
Large Classes ◽  
Infinite Dimensional ◽  
Hyponormal Operators ◽  
Scalar Operator ◽  
Hyperinvariant Subspaces

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.

scholarly journals A Hilbert-Schmidt norm inequality associated with the Fuglede-Putnam theorem

1982 ◽  
Vol 25 (2) ◽  
pp. 177-185 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Norm Inequality ◽  
Hyponormal Operator ◽  
Hyponormal Operators ◽  
Schmidt Norm ◽  
Schmidt Operator

The familiar Fuglede-Putnam theorem asserts that AX = XB implies A*X = XB* when A and B are normal. We prove that A and B* be hyponormal operators and let C be a hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert-Schmidt operator X, the Hilbert-Schmidt norm of AXD + CXB is greater than or equal to the Hilbert-Schmidt norm of A*XD* + C*XB*. In particular, AXD = CXB implies A*XD* = C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert-Schmidt operator of the hypothesis on X to be every operator and still retain the inequality under some suitable hypotheses.

Another Version of Fuglede–Putnam Theorem

2009 ◽  
Vol 16 (3) ◽  
pp. 427-433
Author(s):  
Aissa Nasli Bakir ◽  
Salah Mecheri
Keyword(s):  
Hyponormal Operator ◽  
Hyponormal Operators ◽  
Dominant Operator

Abstract In [Yoshino, Proc. Amer. Math. Soc. 95: 571–572, 1985] the author proved that for a 𝑀-hyponormal operator 𝐴* and for a dominant operator 𝐵, 𝐶𝐴 = 𝐵𝐶 implies 𝐶𝐴* = 𝐵*𝐶. In the case where 𝐴* and 𝐵 are normal, this result is known as the Fuglede–Putnam theorem. In this paper, we will extend this result to the case in which 𝐴 is an injective (𝑝, 𝑘)-quasihyponormal operator and 𝐵* is a dominant operator. We also show that the same result remains valid for (𝑝, 𝑘)-quasihyponormal and log-hyponormal operators.

scholarly journals Hyponormal operators on uniformly smooth spaces

1991 ◽  
Vol 50 (1) ◽  
pp. 150-159
Author(s):  
Muneo Chō
Keyword(s):  
Hyponormal Operator ◽  
Joint Spectrum ◽  
Uniformly Smooth ◽  
Hyponormal Operators ◽  
Smooth Space

AbstractIn this paper we will characterize the spectrum of a hyponormal operator and the joint spectrum of a doubly commuting n-tuple of strongly hyponormal operators on a uniformly smooth space. We also describe some applications of these results.

scholarly journals Weyl type theorem for operator matrices

2008 ◽  
Vol 186 (1) ◽  
pp. 29-39
Author(s):  
Xiaohong Cao
Keyword(s):  
Type Theorem ◽  
Operator Matrices

scholarly journals Generalized Weyl's theorem and hyponormal operators

2004 ◽  
Vol 76 (2) ◽  
pp. 291-302 ◽  
Author(s):  
M. Berkani ◽  
A. Arroud
Keyword(s):  
Hilbert Space ◽  
Weyl’S Theorem ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Isolated Eigenvalues

AbstractLet T be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of T is the set σBW(T) of all λ ∈ Сsuch that T − λI is not a B-Fredholm operator of index 0. Let E(T) be the set of all isolated eigenvalues of T. The aim of this paper is to show that if T is a hyponormal operator, then T satisfies generalized Weyl's theorem σBW(T) = σ(T)/E(T), and the B-Weyl spectrum σBW(T) of T satisfies the spectral mapping theorem. We also consider commuting finite rank perturbations of operators satisfying generalized Weyl's theorem.

scholarly journals Some remarks on the invariant subspace problem for hyponormal operators

2001 ◽  
Vol 28 (6) ◽  
pp. 359-365
Author(s):  
Vasile Lauric
Keyword(s):  
Invariant Subspace ◽  
Invariant Subspaces ◽  
Hyponormal Operator ◽  
Hyponormal Operators ◽  
Subspace Problem ◽  
Invariant Subspace Problem

We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators.

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