- Banach - mathdept.okstate.edu

Abstract of a paper by Venta Terauds
by alspach＠fourier.math.okstate.edu 30 Apr '08

30 Apr '08

This is an announcement for the paper "Functional calculus extensions on dual spaces" by Venta Terauds. Abstract: In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space, the classes of finitely spectral and prespectral operators coincide. We also apply this result to give some sufficient conditions for an operator with an absolutely continuous functional calculus to admit a bounded Borel one. Archive classification: math.FA Mathematics Subject Classification: 47B40 Remarks: 7 pages The source file(s), func_calc_extns_terauds.tex: 24129 bytes, is(are) stored in gzipped form as 0804.3451.gz with size 7kb. The corresponding postcript file has gzipped size 70kb. Submitted from: venta.terauds(a)newcastle.edu.au The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.3451 or http://arXiv.org/abs/0804.3451 or by email in unzipped form by transmitting an empty message with subject line uget 0804.3451 or in gzipped form by using subject line get 0804.3451 to: math(a)arXiv.org.

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Abstract of a paper by Mark Veraar and Tuomas Hytonen
by alspach＠fourier.math.okstate.edu 30 Apr '08

30 Apr '08

This is an announcement for the paper "R-boundedness of smooth operator-valued functions" by Mark Veraar and Tuomas Hytonen. Abstract: In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. Archive classification: math.FA math.PR Mathematics Subject Classification: 47B99; 46B09; 46E35; 46E40 The source file(s), rboundedsmooth_arxiv.tex: 81153 bytes, is(are) stored in gzipped form as 0804.3313.gz with size 24kb. The corresponding postcript file has gzipped size 142kb. Submitted from: mark(a)profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.3313 or http://arXiv.org/abs/0804.3313 or by email in unzipped form by transmitting an empty message with subject line uget 0804.3313 or in gzipped form by using subject line get 0804.3313 to: math(a)arXiv.org.

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Abstract of a paper by Dusan Repovs and Pavel V. Semenov
by alspach＠fourier.math.okstate.edu 30 Apr '08

30 Apr '08

This is an announcement for the paper "A unified construction yielding precisely Hilbert and James sequences spaces" by Dusan Repovs and Pavel V. Semenov. Abstract: Following James' approach, we shall define the Banach space $J(e)$ for each vector $e=(e_1,e_2,...,e_d) \in \Bbb{R}^d$ with $ e_1 \ne 0$. The construction immediately implies that $J(1)$ coincides with the Hilbert space $i_2$ and that $J(1;-1)$ coincides with the celebrated quasireflexive James space $J$. The results of this paper show that, up to an isomorphism, there are only the following two possibilities: (i) either $J(e)$ is isomorphic to $l_2$ ,if $e_1+e_2+...+e_d\ne 0$ (ii) or $J(e)$ is isomorphic to $J$. Such a dichotomy also holds for every separable Orlicz sequence space $l_M$. Archive classification: math.GN math.FA Mathematics Subject Classification: 54C60; 54C65; 41A65; 54C55; 54C20 The source file(s), ArchiveVersion.tex: 21648 bytes, is(are) stored in gzipped form as 0804.3131.gz with size 8kb. The corresponding postcript file has gzipped size 65kb. Submitted from: dusan.repovs(a)guest.arnes.si The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.3131 or http://arXiv.org/abs/0804.3131 or by email in unzipped form by transmitting an empty message with subject line uget 0804.3131 or in gzipped form by using subject line get 0804.3131 to: math(a)arXiv.org.

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Abstract of a paper by Hirbod Assa
by alspach＠fourier.math.okstate.edu 18 Apr '08

18 Apr '08

This is an announcement for the paper "Characterization of compact subsets of $\mathcal{A}^p$ with respect to weak topology" by Hirbod Assa. Abstract: In this brief article we characterize the relatively compact subsets of $\mathcal{A}^p$ for the topology $\sigma(\mathcal{A}^p,\mathcal{R}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces $\mathcal{R}^q$ endowed with the weak topology induced by $\mathcal{A}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of $\mathcal{A}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on $\mathcal{R}^q$ and for their use in the optimization theory. Archive classification: math.PR math.FA Remarks: 8 pages The source file(s), compactsetsAssa.H.tex: 19008 bytes, is(are) stored in gzipped form as 0804.2873.gz with size 6kb. The corresponding postcript file has gzipped size 67kb. Submitted from: assa(a)dms.umontreal.ca The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.2873 or http://arXiv.org/abs/0804.2873 or by email in unzipped form by transmitting an empty message with subject line uget 0804.2873 or in gzipped form by using subject line get 0804.2873 to: math(a)arXiv.org.

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Abstract of a paper by Boris Burshteyn
by alspach＠fourier.math.okstate.edu 18 Apr '08

18 Apr '08

This is an announcement for the paper "Uniform lamda-adjustment and mu-approximation in Banach spaces" by Boris Burshteyn. Abstract: We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fr\'{e}chet-Urysohn in weak* topology. We also introduce another concept of perturbation called uniform mu-approximation which is weaker than perturbations by small gap, norm, and compact convergence, yet stronger than uniform lambda-adjustment. We present Fredholm stability theorems for uniform mu-approximation in arbitrary Banach spaces and a theorem on stability of Riesz kernels and ranges for commuting closed essentially Kato operators. Finally, we define the new concepts of a tuple of subspaces and of a complex of subspaces in Banach spaces, and present stability theorems for index and defect numbers of Fredholm tuples and complexes under uniform lambda-adjustment and uniform mu-approximation. Archive classification: math.FA Mathematics Subject Classification: 32A70; 46A32; 46B50; 47A53; 47A55; 47B07; Remarks: 90 pages The source file(s), boris997paper1.tex: 300886 bytes (looks big), is(are) stored in gzipped form as 0804.2832.gz with size 63kb. The corresponding postcript file has gzipped size 446kb. Submitted from: boris997(a)astound.net The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.2832 or http://arXiv.org/abs/0804.2832 or by email in unzipped form by transmitting an empty message with subject line uget 0804.2832 or in gzipped form by using subject line get 0804.2832 to: math(a)arXiv.org.

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Abstract of a paper by Ostrovsky E. Sirota L
by alspach＠fourier.math.okstate.edu 18 Apr '08

18 Apr '08

This is an announcement for the paper "Nikol'skii-type inequalities for rearrangement invariant spaces" by Ostrovsky E. Sirota L. Abstract: In this paper we generalize the classical Nikol'skii inequality on the many popular classes pairs of rearrangement invariant (r.i.) spaces and construct some examples in order to show the exactness of our estimations. Archive classification: math.FA Mathematics Subject Classification: 60G17 The source file(s), Nik14_4.tex: 40903 bytes, is(are) stored in gzipped form as 0804.2311.gz with size 14kb. The corresponding postcript file has gzipped size 85kb. Submitted from: leos(a)post.sce.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.2311 or http://arXiv.org/abs/0804.2311 or by email in unzipped form by transmitting an empty message with subject line uget 0804.2311 or in gzipped form by using subject line get 0804.2311 to: math(a)arXiv.org.

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Abstract of a paper by Dmitry B. Rokhlin
by alspach＠fourier.math.okstate.edu 15 Apr '08

15 Apr '08

This is an announcement for the paper "Kreps-Yan theorem for Banach ideal spaces" by Dmitry B. Rokhlin. Abstract: Let $C$ be a closed convex cone in a Banach ideal space $X$ on a measurable space with a $\sigma$-finite measure. We prove that conditions $C\cap X_+=\{0\}$ and $C\supset -X_+$ imply the existence of a strictly positive continuous functional on $X$, whose restriction to $C$ is non-positive. Archive classification: math.FA Mathematics Subject Classification: 46E30; 46B42 Remarks: 6 pages The source file(s), RokhlinKreps-Yantheoremforbanachidealspaceseng.tex: 18929 bytes, is(are) stored in gzipped form as 0804.2075.gz with size 7kb. The corresponding postcript file has gzipped size 73kb. Submitted from: rokhlin(a)math.rsu.ru The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.2075 or http://arXiv.org/abs/0804.2075 or by email in unzipped form by transmitting an empty message with subject line uget 0804.2075 or in gzipped form by using subject line get 0804.2075 to: math(a)arXiv.org.

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Abstract of a paper by Ariel Blanco and Niels Groenbaek
by alspach＠fourier.math.okstate.edu 14 Apr '08

14 Apr '08

This is an announcement for the paper "Amenability of algebras of approximable operators" by Ariel Blanco and Niels Groenbaek. Abstract: We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space. Archive classification: math.FA Mathematics Subject Classification: 46B20, 47L10 (primary), 16E40 (secondary) Remarks: 20 pages, to appear in Israel Journal of Mathematics The source file(s), OnAmenability2.tex: 82733 bytes, is(are) stored in gzipped form as 0804.1725.gz with size 25kb. The corresponding postcript file has gzipped size 148kb. Submitted from: gronbaek(a)math.ku.dk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.1725 or http://arXiv.org/abs/0804.1725 or by email in unzipped form by transmitting an empty message with subject line uget 0804.1725 or in gzipped form by using subject line get 0804.1725 to: math(a)arXiv.org.

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Abstract of a paper by William Arveson
by alspach＠fourier.math.okstate.edu 10 Apr '08

10 Apr '08

This is an announcement for the paper "Maximal vectors in Hilbert space and quantum entanglement" by William Arveson. Abstract: Given two matrix algebras $M_1$, $M_2$, the natural inclusion of $\mathcal L^1(M_1\otimes M_2)$ in the projective tensor product of Banach spaces $\mathcal L^1(M_1)\hat\otimes \mathcal L^1(M_2)$ is a contraction but not an isometry; and the projective cross norm can be restricted to the convex set $\mathcal S$ of density matrices in $M_1\otimes M_2$to obtain a continuous convex function $E:\mathcal S\to [1,\infty)$. We show that $E$ {\em faithfully measures entanglement} in the sense that a state is entangled if and only if its density matrix $A$ satisfies $E(A)>1$. Moreover, $E(A)$ is maximized at the density matrix $A$ associated with a pure state if and only if the range of $A$ is generated by a maximally entangled unit vector. These concrete results follow from a general analysis of norm-closed subsets $V$ of the unit sphere of a Hilbert space $H$. A {\em maximal vector} (for $V$) is a unit vector $\xi\in H$ whose distance to $V$ is maximum. Maximal vectors generalize the ``maximally entangled" unit vectors of quantum theory. In general, under a mild regularity hypothesis on $V$ we show that there is a {\em norm} on $\mathcal L^1(H)$ whose restriction to the convex set $\mathcal S$ of density operators achieves its minimum precisely on the closed convex hull of the rank one projections associated with vectors in $V$. It achieves its maximum on a rank one projection precisely when its unit vector is a maximal vector. This ``entanglement-measuring norm" is unique, and computation shows it to be the projective cross norm in the above setting of bipartite tensor products $H=H_1\otimes H_2$. Archive classification: math.OA math.FA Mathematics Subject Classification: 46N50,81P68, 94B27 Remarks: 25 pages The source file(s), ent4.tex: 76983 bytes, is(are) stored in gzipped form as 0804.1140.gz with size 21kb. The corresponding postcript file has gzipped size 126kb. Submitted from: arveson(a)math.berkeley.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.1140 or http://arXiv.org/abs/0804.1140 or by email in unzipped form by transmitting an empty message with subject line uget 0804.1140 or in gzipped form by using subject line get 0804.1140 to: math(a)arXiv.org.

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Abstract of a paper by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis
by alspach＠fourier.math.okstate.edu 10 Apr '08

10 Apr '08

This is an announcement for the paper "Stochastic evolution equations in UMD Banach spaces" by J.M.A.M. van Neerven, M.C. Veraar, and L. Weis. Abstract: We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation \[\left\{\begin{aligned} dU(t) & = (AU(t) + F(t,U(t)))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,\Tend],\\ U(0) & = u_0, \end{aligned} \right. \] where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^\l([0,T];\D((-A)^\theta)$ provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy $\l+\theta<\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations. Archive classification: math.FA math.PR Mathematics Subject Classification: 47D06; 60H15; 28C20; 46B09 Remarks: Accepted for publication in Journal of Functional Analysis The source file(s), scp_arxiv.tex: 157532 bytes, is(are) stored in gzipped form as 0804.0932.gz with size 44kb. The corresponding postcript file has gzipped size 241kb. Submitted from: mark(a)profsonline.nl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.0932 or http://arXiv.org/abs/0804.0932 or by email in unzipped form by transmitting an empty message with subject line uget 0804.0932 or in gzipped form by using subject line get 0804.0932 to: math(a)arXiv.org.

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