scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

scholarly journals Expansion of analytic functions of an operator in series of Faber polynomials

1997 ◽  
Vol 56 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Analytic Functions ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Faber Polynomials ◽  
Bounded Linear ◽  
In Series ◽  
Image Position

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined bywhere C is a contour surrounding SP(T) and contained in D.

Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
Author(s):  
Fredric M. Pollack
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Subset ◽  
Bounded Linear Operator ◽  
Convex Sets ◽  
Numerical Range ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Empty Convex ◽  
Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

scholarly journals On generalization of decomposability

1981 ◽  
Vol 22 (1) ◽  
pp. 77-81 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Bounded Linear Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Open Cover ◽  
Bounded Linear ◽  
Image Position ◽  
Finite Open Cover

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

A Spectral Theory for Duality Systems of Operators on a Banach Space

1970 ◽  
Vol 22 (5) ◽  
pp. 994-996 ◽  
Author(s):  
J. G. Stampfli
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Banach Theorem ◽  
Duality Map ◽  
Image Position

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.A duality system for the operator T is an ordered sextuple(i) T is a bounded linear operator mapping the Banach space B into B,(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1010-1030 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Growth Conditions ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

scholarly journals Support projections on Banach spaces

1977 ◽  
Vol 18 (1) ◽  
pp. 13-15 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Support Projection

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.

scholarly journals Some convexity theorems for matrices

1971 ◽  
Vol 12 (2) ◽  
pp. 110-117 ◽  
Author(s):  
P. A. Fillmore ◽  
J. P. Williams
Keyword(s):  
Linear Operator ◽  
Operator Theory ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Two Dimensional ◽  
Bounded Linear ◽  
Image Position

The numerical range of a bounded linear operator A on a complex Hilbertspace H is the set W(A) = {(Af, f): ║f║ = 1}. Because it is convex andits closure contains the spectrum of A, the numerical range is often a useful toolin operator theory. However, even when H is two-dimensional, the numerical range of an operator can be large relative to its spectrum, so that knowledge of W(A) generally permits only crude information about A. P. R. Halmos [2] has suggested a refinement of the notion of numerical range by introducing the k-numerical rangesfor k = 1, 2, 3, …. It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

scholarly journals Perturbations of local C-cosine functions

2020 ◽  
Vol 65 (4) ◽  
pp. 585-597
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Bounded Linear ◽  
Cosine Functions

"We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by $$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$ for all $x\in\tX$ and $0\leq t<T_0$, if $\tA$ is a closed subgenerator of a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds: $(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with $\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with $\tC(\cdot)$ on $\tX$. Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and $$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$ for all $x\in\tX$ and $0\leq t<T_0$."

scholarly journals Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

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