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2004, Journal of mathematical analysis …
In this note, we characterize nice operators in a class of Banach spaces, which includes spaces C(K) and L 1 (µ), as those operators that preserve extreme points.
Banach Journal of Mathematical Analysis, 2016
The aim of this paper is to describe the operators between spaces of continuously differentiable functions whose adjoint preserves extreme points. It is important to mention that no condition regarding injectivity or surjectivity of the operators is assumed. Previously known results characterizing surjective isometries can be immediately derived from such descriptions.
Integral Equations and Operator Theory, 2014
Let M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace τ and E be a strongly symmetric Banach function space on [0, τ (1)). We show that an operator x in the unit sphere of E(M, τ ) is k-extreme, k ∈ N, whenever its singular value function µ(x) is k-extreme and one of the following conditions hold (i) µ(∞, x) = limt→∞ µ(t, x) = 0 or (ii) n(x)Mn(x * ) = 0 and |x| ≥ µ(∞, x)s(x), where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever M is nonatomic. The global k-rotundity property follows, that is if M is non-atomic then E is k-rotund if and only if E(M, τ ) is k-rotund. As a consequence of the noncommutive results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement µ(f ) is k-extreme and |f | ≥ µ(∞, f ). We conclude with the corollary on orbits Ω(g) and Ω ′ (g). We get that f is a k-extreme point of the orbit Ω(g), g ∈ L 1 + L∞, or Ω ′ (g), g ∈ L 1 [0, α), α < ∞, if and only if µ(f ) = µ(g) and |f | ≥ µ(∞, f ). From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.
Miskolc Mathematical Notes, 2015
In this paper we show that on complete metric spaces the class of weakly Picard operators contains some operators which are more general than the class of almost contractions.
Proceedings of the Edinburgh Mathematical Society, 1972
The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.
2004
Using the notion of a Banach operator, we have obtained a decompositional property of a Hilbert space, and the equality of two invertible bounded linear multiplicative operators on a normed algebra with identity.
IOSR Journals , 2019
This paper shows that a completely continuous operator is continuous but every continuous operator is not completely continuous whereas continuous operator of finite rank is completely continuous operator.
2017
First, we give some properties of class A(k) operators which are defined in [21]. Exactly we show that if Tn is a compact operator from class A(k), then it follows that T is compact too. We introduce the class M − A(k) of operators and show some properties of this class of operators and their relationship with other classes. We also prove that if Pμ is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a class A(k) operator T where k ∈ (0, 1], then Pμ is self-adjoint, if and only if, ker(T −μ) ⊆ ker(T ∗ − μ). Second, we prove that a class A(k) of operator, where k ∈ (0, 1], is polaroid and a−Weyl’s theorem holds for class A(k) of operators. Finally, we see if T is a contraction of the class A(k) operator for k > 0, then the nonnegative operator D = ( T ∗|T |T ) 1 k+1 − |T | is a contraction whose power sequence {D}n=1 convergences strongly to a projection P and TP = O. Also, we prove if T ∈ A(k) is a contraction for 0 < k ≤ 1, then T has a Wold−type decompo...
2008
First of all, I would like to thank my supervisor, Professor Richard Haydon, for suggesting interesting problems to work on and for his constant help and encouragement. I am also immensely grateful to my husband Waldemar for his support throughout my studies. Furthermore, I would like to mention my friends and family for their significant contribution to the non-mathematical aspect of my life. Similarly, special thanks should go to the people who have offered me various employment opportunities thus giving me a chance to develop skills used both in mathematical and non-mathematical environments. This includes teaching positions at St. Hilda's, New, Brasenose and Jesus colleges and the L A T E X job for the academic administration team of the Mathematical Institute.
Topology and its Applications, 2013
Using a strengthening of the concept of K σδ set, introduced in this paper, we study a certain subclass of the class of K σδ Banach spaces; the so called strongly K σδ Banach spaces. This class of spaces includes subspaces of strongly weakly compactly generated (SWCG) as well as Polish Banach spaces and it is related to strongly weakly Kanalytic (SWKA) Banach spaces as the known classes of K σδ and weakly K-analytic (WKA) Banach spaces are related.
Proceedings of The Edinburgh Mathematical Society, 1998
We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = oo. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMO X . This turns out to depend strongly on the convexity of the Banach lattice X. We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.
Journal of Approximation Theory, 2010
We generalize a recent result of de la Cal and Cárcamo concerning an extremal property of Bernstein operators.
Proceedings of the American Mathematical Society, 1972
If H is the Hilbert transform on LP(Z), then T=ttI+íH is a well-bounded operator for \<p<ao, but is not a scalar-type spectral operator except when/>=2. The purpose of this note is to show that there is a well-bounded operator on a reflexive Banach space which is not scalar-type spectral.
2013
Abstract. We introduce a chain condition (B), defined for operators acting on C(K)spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pelczyński’s characterisation of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space then, for an operator T: C(K) → X, T is weakly compact if and only if T satisfies (B) if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal’s lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies (B), for example both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying (B) forms a clos...
Publicacions Matematiques, 2010
We consider maximal operators $M_{\mathcal B}$ with respect to a basis ${\mathcal B}$. In the case when $M_{\mathcal B}$ satisfies a reversed weak type inequality, we obtain a boundedness criterion for $M_{\mathcal B}$ on an arbitrary quasi-Banach function space $X$. Being applied to specific ${\mathcal B}$ and $X$ this criterion yields new and short proofs of a number of well-known
Proceedings of The American Mathematical Society, 2008
We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W 1,p (R) × W 1,q (R) → W 1,r (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R n when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versions.
2008
List of included articles This dissertation consists of an introductory part and the following publications: [A] H. Luiro, Continuity of the maximal operator in Sobolev spaces. Proc. of AMS., 135 (2007), no.1, 243-251. [B] H. Luiro, The regularity of the Hardy-Littlewood maximal operator on subdomains of R n. Proceedings of the Edinburgh Mathematical Society, to appear.
Journal of Operator Theory, 2016
We study extreme points of the unit ball of an operator space by introducing the new notion “(approximate) quasi-identities”. More specifically, we characterize an operator algebra having a contractive (approximate) quasi– (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points of the unit ball (of the weak*-closure) of the underlying operator space. Furthermore, we give a necessary and sufficient condition for a given operator space to be qualified to become a C*-algebra or a one-sided ideal in a C*-algebra in terms of quasi-multipliers.
Mathematics
We characterize the extreme points of the closed unit ball of the dual of a Banach space which are preserved by the adjoint of any extreme operator. The result is related to the structure topology introduced by Alfsen and Effros on the set of all extreme points in the dual of any Banach space. As a consequence, we prove that c0(I) is the only Banach space such that the adjoint of every extreme operator taking values into it preserves extreme points.
Czechoslovak Mathematical Journal, 2010
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2004
This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra or a one-sided ideal in a $C^*$-algebra in terms of quasi-multipliers. An extreme point is also used to show that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra.
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