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. Math 1961 11 9 23 Dijksma, A. and de Snoo, H. S. V. , Self-adjoint extensions

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] Sebestyén , Z. , Kapos , L. 1995 On range characterization of adjoint operators on Hilbert space Studia Sci. Math. Hungar. 30 261 – 263 .

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Arens, R. , The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 13 ,659f Arens R. The

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Abstract

Filters are a fundamental tool in the study of convergence and completeness in topology. On the other hand, downsets have been used extensively for this purpose in the setting of pointfree topology. This paper investigates links between these in the asymmetric context, that is, for biframes and bispaces.

We present an appropriate kind of filter for the asymmetric setting, which we call a bifilter. These form a bispace, functorially so, which proves isomorphic to the spectrum of the downset biframe. As a corollary, downset biframes are seen to be isomorphic to the opens of the bifilter bispace. Both these correspondences are natural isomorphisms.

The join map from a downset biframe to its underlying biframe appears here as a universal strict quotient. We use it to show that the embedding of any T 0 bispace in its bifilter bispace is a universal strict extension.

Banaschewski and Hong [10] have established the importance of general filters for convergence and completeness in the pointfree setting. We conclude this paper with a discussion of an appropriate concept of general bifilter, and show that the right adjoint of the join map mentioned above is a universal general bifilter.

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Abstract  

Imaginary powers associated to the Laguerre differential operator
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are investigated. It is proved that for every multi-index α = (α1,...αd) such that αi ≧ −1/2, αi ∉ (−1/2, 1/2), the imaginary powers
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, of a self-adjoint extension of L α, are Calderón-Zygmund operators. Consequently, mapping properties of
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follow by the general theory.
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. [15] Krein , M. 1947 The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I Math. Sbornik N.S. 20

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[1] Belton , A. C. R ., Operator Theory , Fourth-year undergraduate course , Lancaster University 2018 . [2] Bernau , S. J ., The square root of a positive self-adjoint operator , J. Austr. Math. Soc ., 8 ( 1968 ), 17 – 36 . [3] Bishop

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Abstract  

The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let
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(H) be the algebra of all bounded linear operators on H, and let
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(H) ⊂
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(H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T:
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(H) →
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(H) is a linear mapping satisfying T(AA* A) = T(A)A* AAT(A*)A + AA*T(A) for all A
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(H). Then T is of the form T(A) = AB + BA for all A
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(H), where B is a fixed operator from
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(H). A result concerning functional equations related to bicircular projections is proved
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), 591 – 603 . 9 B irkenmeier , G. F. and P ark , J. K. , Self-adjoint ideals in Baer *-ring , Comm. Algebra , 28 ( 2000 ), 4259 – 4268 . 10

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Summary  

It follows from [1], [4] and [7] that any closed \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n$ \end{document}-codimensional subspace (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n \ge 1$ \end{document} integer) of a real Banach space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X$ \end{document} is the kernel of a projection \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X \to X$ \end{document}, of norm less than \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n) + \varepsilon$ \end{document}~(\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\varepsilon > 0$ \end{document} arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n) < \sqrt{n}$ \end{document} for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n > 1$ \end{document}, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt{n}$ \end{document} rather than \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $f(n)$ \end{document}, has been proved in [2]. A~small improvement of the statement of [2], for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n = 2$ \end{document}, is given in [3], pp.~61--62, Remark.) In [1] for this theorem a deeper statement is used, on approximations of finite rank projections on the dual space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X^*$ \end{document} by adjoints of finite rank projections on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $X$ \end{document}. In this paper we show that the first cited result is an immediate consequence of the principle of local reflexivity, and of the result from [7].

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