Remarks on Invariant Subspace Lattices

1969 ◽  
Vol 12 (5) ◽  
pp. 639-643 ◽  
Author(s):  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Complete Lattice ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Subspace Lattices

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

scholarly journals Invariant subspace lattices of Lambert's weighted shifts

1982 ◽  
Vol 33 (1) ◽  
pp. 135-142
Author(s):  
B. S. Yadav ◽  
S. Chatterjee
Keyword(s):  
Hilbert Space ◽  
Invariant Subspace ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weighted Shifts ◽  
Positive Real ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

scholarly journals An operator is a product of two quasi-nilpotent operators if and only if it is not semi-Fredholm

2006 ◽  
Vol 136 (5) ◽  
pp. 935-944 ◽  
Author(s):  
Roman Drnovšek ◽  
Nika Novak ◽  
Vladimir Müller
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Nilpotent Operators

We prove that a (bounded, linear) operator acting on an infinite-dimensional, separable, complex Hilbert space can be written as a product of two quasi-nilpotent operators if and only if it is not a semi-Fredholm operator. This solves the problem posed by Fong and Sourour in 1984. We also consider some closely related questions. In particular, we show that an operator can be expressed as a product of two nilpotent operators if and only if its kernel and co-kernel are both infinite dimensional. This answers the question implicitly posed by Wu in 1989.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

scholarly journals Spectral radius inequalities for functions of operators defined by power series

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2847-2856
Author(s):  
S.S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Linear Operator ◽  
Spectral Radius ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Commuting Operators ◽  
Functions Of Operators ◽  
The Absolute

By the help of power series f(z)=??,n=0 anzn we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f , namely fa(z):= ??,n=0 |an|zn. Utilising these functions we show among others that r[f(T)] ? fa [r(T)] where r (T) denotes the spectral radius of the bounded linear operator T on a complex Hilbert space while ||T|| is its norm. When we have A and B two commuting operators, then r2[f(AB)]? fa(r2(A)) fa(r2(B)) and r[f(AB)]?1/2[fa(||AB||)+fa(||A2||1/2||B2||1/2)].

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 Generalized Fuglede-Putnam Theorem and $m$-quasi-class $A(k)$ operators

2019 ◽  
pp. 73
Author(s):  
Mohammad H.M. Rashid
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Operator Equation ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Dense Range ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Class A ◽  
Dimensional Hilbert Space

For a bounded linear operator $T$ acting on acomplex infinite dimensional Hilbert space $\h,$ we say that $T$is $m$-quasi-class $A(k)$ operator for $k>0$ and $m$ is apositive integer (abbreviation $T\in\QAkm$) if$T^{*m}\left((T^*|T|^{2k}T)^{\frac{1}{k+1}}-|T|^2\right)T^m\geq0.$ The famous {\it Fuglede-Putnam theorem} asserts that: the operator equation$AX=XB$ implies $A^*X=XB^*$ when $A$ and $B$ are normal operators.In this paper, we prove that if $T\in \QAkm$ and $S^*$ isan operator of class $A(k)$ for $k>0$. Then $TX=XS$, where $X\in\bh$ is an injective with dense range implies $XT^*=S^*X$.

scholarly journals Remarks on n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5441-5451 ◽  
Author(s):  
Muneo Chō ◽  
Ji Lee ◽  
Kotaro Tanahashi ◽  
Atsushi Uchiyama
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Normal Operators

Let T be a bounded linear operator on a complex Hilbert space and n,m ? N. Then T is said to be n-normal if T+Tn = TnT+ and (n,m)-normal if T+mTn = TnT+m. In this paper, we study several properties of n-normal, (n,m)-normal operators. In particular, we prove that if T is 2-normal with ?(T) ? (-?(T)) ? {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n,m)-normal, then T is decomposable and Weyl?s theorem holds for f (T), where f is an analytic function on ?(T) which is not constant on each of the components of its domain.

scholarly journals Factoring a Quadratic Operator as a Product of Two Positive Contractions

2016 ◽  
Vol 59 (2) ◽  
pp. 354-362 ◽  
Author(s):  
Chi-Kwong Li ◽  
Ming-Cheng Tsai
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Positive Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Necessary Condition ◽  
Operator Matrix ◽  
Quadratic Operator ◽  
Bounded Linear

AbstractLet T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the formfor some a, b ∊ [0, 1] and strictly positive operator P with . Also, we give a necessary condition for a bounded linear operator T with operator matrix on H⊕K that can be written as a product of two positive contractions.

Products of positive operators

1974 ◽  
Vol 76 (2) ◽  
pp. 415-416 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Positive Operators ◽  
Bounded Linear

Let H be a complex Hilbert space. Recall that a bounded linear operator A, on H, is positive if (Ax, x) ≥ 0 (x ∈ H) (so that A = A* necessarily) and positive definite if A is positive and invertible.

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