scholarly journals Points of narrowness and uniformly narrow operators

2017 ◽  
Vol 9 (1) ◽  
pp. 37-47 ◽  
Author(s):  
A.I. Gumenchuk ◽  
I.V. Krasikova ◽  
M.M. Popov
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Measure Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Standard Tool ◽  
Orthogonally Additive ◽  
Orthogonally Additive Operators ◽  
Continuous Linear Operators

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.

An operator version of Abel's continuity theorem

2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Continuous Linear ◽  
Continuity Theorem ◽  
Operator Version ◽  
Continuous Linear Operators

Abstract For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

1989 ◽  
Vol 32 (4) ◽  
pp. 450-458
Author(s):  
Takemitsu Kiyosawa
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergent Sequence ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Infinite Dimensional ◽  
Valued Field ◽  
Infinite Dimensional Banach Space ◽  
Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

scholarly journals Integration in Orlicz-Bochner Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Marian Nowak
Keyword(s):  
Integral Representation ◽  
Measure Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Continuous Linear ◽  
Topological Properties ◽  
Young Function ◽  
Finite Measure Space ◽  
Bochner Space ◽  
Continuous Linear Operators

Let (Ω,Σ,μ) be a complete σ-finite measure space, φ be a Young function, and X and Y be Banach spaces. Let Lφ(X) denote the Orlicz-Bochner space, and Tφ∧ denote the finest Lebesgue topology on Lφ(X). We study the problem of integral representation of (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y with respect to the representing operator-valued measures. The relationships between (Tφ∧,·Y)-continuous linear operators T:Lφ(X)→Y and the topological properties of their representing operator measures are established.

Continuous linear operators on C(K, X) and pointwise weakly precompact subsets of C(K, X)

1992 ◽  
Vol 111 (1) ◽  
pp. 143-150 ◽  
Author(s):  
A. lger
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Supremum Norm ◽  
Continuous Linear ◽  
Continuous Linear Operators

AbstractLet K be a compact Hausdorif space, X a Banach space and C(K, X) the Banach space of all continuous functions : KX equipped with the supremum norm. A subset H of C(K, X) is pointwise weakly precompact if, for each t in K, the set Ht) = {(t):H} is weakly precompact. In this note we study the images of a bounded pointwise weakly precompact subset H of C(K, X) under several classes of linear operators on C(K, X).

scholarly journals Commutation Properties of Operator Polynomials

1971 ◽  
Vol 12 (1) ◽  
pp. 98-100
Author(s):  
S. R. Caradus
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Reverse Implication ◽  
Do So ◽  
Continuous Linear Operators

Suppose A and B are continuous linear operators mapping a complex Banach space X into itself. For any polynomial pC, it is obvious that when A commutes with B, then p(A) commutes with B. To see that the reverse implication is false, let A be nilpotent of order n. Then An commutes with all B but A cannot do so. Sufficient conditions for the implication: p(A) commutes with B implies A commutes with B: were given by Embry [2] for the case p(λ) = λn and Finkelstein and Lebow [3] in the general case. The latter authors proved in fact that if f is a function holomorphic on σ(A) and if f is univalent with non-vanishing derivative on σ(A), then A can be expressed as a function of f(A).

Lomonosov's hyperinvariant subspace theorem for real spaces

1981 ◽  
Vol 89 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. D. Hooker
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Linear Operators ◽  
Continuous Linear ◽  
Hyperinvariant Subspace ◽  
Subspace Theorem ◽  
Hyperinvariant Subspaces ◽  
A New Technique ◽  
Continuous Linear Operators

In 1973, V.I.Lomonosov introduced a new technique for finding invariant and hyperinvariant subspaces for certain classes of (continuous, linear) operators on complex Banach spaces. Recall that a closed subspace M of the Banach space X is called hyperinvariant for the operator T if S(M) ⊂ M for every operator S which commutes with T.

Copies of l∞ in an operator space

1990 ◽  
Vol 108 (3) ◽  
pp. 523-526 ◽  
Author(s):  
Lech Drewnowski
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Operator Space ◽  
Isomorphic Copy ◽  
Continuous Linear ◽  
Weakly Continuous ◽  
Continuous Linear Operators

Let X and Y be Banach spaces. Then Kw*(X*, Y) denotes the Banach space of compact and weak*-weakly continuous linear operators from X* into Y, endowed with the usual operator norm. Let us write E⊃l∞ to indicate that a Banach space E contains an isomorphic copy of l∞. The purpose of this note is to prove the followingTheorem. Kw*(X*, Y) ⊃ l∞if and only if either X ⊃ l∞or Y ⊃ l∞.

scholarly journals Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

1988 ◽  
Vol 30 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Volker Wrobel
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
Continuous Linear Operators

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

scholarly journals Modulus hyperinvariant ideals for a finitely quasinilpotent operators

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4717-4720
Author(s):  
Junfeng Liu
Keyword(s):  
Banach Space ◽  
Unconditional Basis ◽  
Linear Operators ◽  
Closed Ideal ◽  
Continuous Linear ◽  
Positive Vector ◽  
Quasinilpotent Operators ◽  
Continuous Linear Operators

Let X be a Banach space with an unconditional basis, and let C ? {0} be a collection of continuous linear operators with modulus on X that is finitely modulus-quasinilpotent at a non-zero positive vector. Then C and its right modulus sub-commutant C'm have a common non-trivial invariant closed ideal.

scholarly journals The non-emptiness of joint spectral subsets of euclidean n-space

1989 ◽  
Vol 47 (2) ◽  
pp. 300-306 ◽  
Author(s):  
W. J. Ricker ◽  
A. R. Schep
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Spectral Set ◽  
Continuous Linear Operators ◽  
Real Euclidean Space

AbstractA.McIntosh and A. Pryde introduced and gave some applications of notion of “spectral set”, γ(T), associated with each finite, commuting family of continuous linear operators T in a Banach space. Unlike most concepts of joint spectrum, the set γ(T) is part of real Euclidean space. It is shown that γ(T) is always non-empty whenver there are at least two operators in T.

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