scholarly journals Some Estimates of Certain Subnormal and Hyponormal Derivations

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Vasile Lauric
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Schmidt Operator ◽  
Norm Ideal ◽  
Class Of Functions

We prove that if and are subnormal operators and is a bounded linear operator such that is a Hilbert-Schmidt operator, then is also a Hilbert-Schmidt operator and for belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that , are hyponormal operators and is such that belongs to a norm ideal , and we prove that and for being in a certain class of functions.

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.

scholarly journals On numerical ranges of generalized derivations and related properties

1984 ◽  
Vol 36 (1) ◽  
pp. 134-142 ◽  
Author(s):  
Sen-Yen Shaw
Keyword(s):  
Numerical Range ◽  
Normal Operator ◽  
Linear Operators ◽  
Minimal Norm ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Normal Operators ◽  
Schmidt Operator ◽  
The Difference ◽  
Norm Ideal

AbstractThis paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.

Fuglede–Putnam type theorems via the generalized second Aluthge transform

2018 ◽  
Vol 13 (01) ◽  
pp. 2050018
Author(s):  
Manzar Maleki ◽  
Ali Reza Janfada
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Polar Decomposition ◽  
Aluthge Transform ◽  
Bounded Linear ◽  
Schmidt Operator

Let [Formula: see text] and [Formula: see text] be the polar decompositions of [Formula: see text] and [Formula: see text]. A pair [Formula: see text] is said to have the FP-property if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. Let [Formula: see text] denote the generalized second Aluthge transform of a bounded linear operator [Formula: see text] such that [Formula: see text] is the polar decomposition of [Formula: see text] where [Formula: see text] denotes the first Aluthge transform of operator [Formula: see text]. We show that (i) if [Formula: see text] is class [Formula: see text] and [Formula: see text] is invertible class [Formula: see text] operators with [Formula: see text] such that [Formula: see text] for some Hilbert Schmidt operator [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] for any [Formula: see text], then [Formula: see text] for any [Formula: see text], furthermore, if [Formula: see text] is invertible, then [Formula: see text]. Finally, if [Formula: see text] and [Formula: see text] and [Formula: see text] is an operator such that [Formula: see text], then we prove that [Formula: see text] for any [Formula: see text] such that [Formula: see text].

scholarly journals Fuzzy Paranormal Operators

2019 ◽  
Vol 9 (1) ◽  
pp. 3123-3127
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unit Vector ◽  
Bounded Linear ◽  
Hyponormal Operators

In this paper, we introduced and discussed about fuzzy paranormal operators. A fuzzy bounded linear operator on a fuzzy Hilbert space is fuzzy paranormal if , for every unit vector a in It is easily known that this class includes fuzzy hyponormal operators.

scholarly journals On relaxation normality in the Fuglede-Putnam theorem for a quasi-class $A$ operators

2009 ◽  
Vol 40 (3) ◽  
pp. 307-312 ◽  
Author(s):  
M. H. M. Rashid ◽  
M. S. M. Noorani
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Class A ◽  
Schmidt Operator

Let $T$ be a bounded linear operator acting on a complex Hilbert space $ \mathcal{H} $. In this paper, we show that if $A$ is quasi-class $A$, $ B^* $ is invertible quasi-class $A$, $X$ is a Hilbert-Schmidt operator, $AX=XB$ and $ \left\Vert |A^*| \right\Vert \left\Vert |B|^{-1} \right\Vert \leq 1 $, then $ A^* X = X B^* $.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals EXPLICIT INTERPOLATION BOUNDS BETWEEN HARDY SPACE AND

2013 ◽  
Vol 95 (2) ◽  
pp. 158-168
Author(s):  
H.-Q. BUI ◽  
R. S. LAUGESEN
Keyword(s):  
Growth Rate ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Mean Oscillation ◽  
Complex Interpolation ◽  
Bounded Linear ◽  
Exponential Bound ◽  
Mean Oscillation ◽  
Good Lambda Inequality ◽  
Interpolation Result

AbstractEvery bounded linear operator that maps ${H}^{1} $ to ${L}^{1} $ and ${L}^{2} $ to ${L}^{2} $ is bounded from ${L}^{p} $ to ${L}^{p} $ for each $p\in (1, 2)$, by a famous interpolation result of Fefferman and Stein. We prove ${L}^{p} $-norm bounds that grow like $O(1/ (p- 1))$ as $p\downarrow 1$. This growth rate is optimal, and improves significantly on the previously known exponential bound $O({2}^{1/ (p- 1)} )$. For $p\in (2, \infty )$, we prove explicit ${L}^{p} $ estimates on each bounded linear operator mapping ${L}^{\infty } $ to bounded mean oscillation ($\mathit{BMO}$) and ${L}^{2} $ to ${L}^{2} $. This $\mathit{BMO}$ interpolation result implies the ${H}^{1} $ result above, by duality. In addition, we obtain stronger results by working with dyadic ${H}^{1} $ and dyadic $\mathit{BMO}$. The proofs proceed by complex interpolation, after we develop an optimal dyadic ‘good lambda’ inequality for the dyadic $\sharp $-maximal operator.

scholarly journals Density and representation theorems for multipliers of type (p, q)

1967 ◽  
Vol 7 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Alessandro Figà-Talamanca ◽  
G. I. Gaudry
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Important Class ◽  
Lebesgue Spaces ◽  
Representation Theorems ◽  
Locally Compact ◽  
Bounded Linear ◽  
Character Group ◽  
Hausdorff Group ◽  
Lca Group

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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