The Range Sequence of an Operator

1974 ◽  
Vol 17 (1) ◽  
pp. 145-147
Author(s):  
F.-H. Vasilescu
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Linear Subspaces ◽  
Image Position

Let T be a linear operator on a Banach space X and consider the sequence of rangeswhere the inclusions are not necessarily proper. The linear subspaces Xn=TnX (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X0=X and denote by |x|0 (x∈X0) the norm of X0.

scholarly journals On well-bounded operators of class Г

1983 ◽  
Vol 24 (1) ◽  
pp. 1-5
Author(s):  
Adnan A. S. Jibril
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Operators ◽  
Strong Limit ◽  
Riemann Sums ◽  
Image Position

Let T be a linear operator acting in a Banach space X. It has been shown by Smart [5] and Ringrose [3] that, if X is reflexive, then T is well-bounded if and only if it may be expressed in the formwhere {E(λ)} is a suitable family of projections in X and the integral exists as the strong limit of Riemann sums.

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.

Hadamard–Landau inequalities in uniformly convex spaces

1981 ◽  
Vol 90 (2) ◽  
pp. 259-264 ◽  
Author(s):  
J. R. Partington
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Uniformly Convex ◽  
Image Position ◽  
Uniformly Convex Spaces ◽  
Convex Spaces

The inequalityfor fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extensionfor T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

An unbounded spectral mapping theorem

1968 ◽  
Vol 8 (1) ◽  
pp. 119-127 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Complex Numbers ◽  
Spectral Mapping Theorem ◽  
Complex Polynomials ◽  
Spectral Mapping ◽  
Image Position ◽  
Densely Defined

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

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).).

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.

Banach Envelopes of Non-Locally Convex Spaces

1986 ◽  
Vol 38 (1) ◽  
pp. 65-86 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Bounded Linear Operator ◽  
Locally Convex Spaces ◽  
Minkowski Functional ◽  
Bounded Linear ◽  
Banach Envelope ◽  
Image Position ◽  
Locally Convex

Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

Idempotent Multipliers of H1(T)

1987 ◽  
Vol 39 (5) ◽  
pp. 1223-1234 ◽  
Author(s):  
I. Klemes
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Image Position

Let as usual T = R/2πZ be the circle, and H1 the subspace of L1(T) of all f such that for all integers n < 0. The normrestricted to H1, makes it a Banach space. By a multiplier of H1 we mean a bounded linear operator m:H1 → H1 such that there is a sequence in C with

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.

On the Stability and Boundedness of Differential Systems

1962 ◽  
Vol 58 (3) ◽  
pp. 492-496 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Real Banach Space ◽  
Closed Linear Operator ◽  
Differential Systems ◽  
The Real ◽  
The Stability ◽  
Definition Of ◽  
Image Position ◽  
The Right

Consider the differential systemswhere A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.

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