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STRUCTURE THEOREM FOR $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{AN}$ -OPERATORS

  • G. RAMESH (a1)

Abstract

In this paper we prove a structure theorem for the class of $\mathcal{AN}$ -operators between separable, complex Hilbert spaces which is similar to that of the singular value decomposition of a compact operator. Apart from this, we show that a bounded operator is $\mathcal{AN}$ if and only if it is either compact or a sum of a compact operator and scalar multiple of an isometry satisfying some condition. We obtain characterizations of these operators as a consequence of this structure theorem and deduce several properties which are similar to those of compact operators.

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References

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[1]Bakić, D., ‘Compact operators, the essential spectrum and the essential numerical range’, Math. Commun. 3(1) (1998), 103108.
[2]Carvajal, X. and Neves, W., ‘Operators that achieve the norm’, Integral Equations Operator Theory 72(2) (2012), 179195.
[3]Conway, J. B., A Course in Functional Analysis, Graduate Texts in Mathematics, 96, 2nd edn (Springer, New York, 1990).
[4]Douglas, R. G., Banach Algebra Techniques in Operator Theory, Pure and Applied Mathematics, 49 (Academic Press, New York, 1972).
[5]Fillmore, P. A., Stampfli, J. G. and Williams, J. P., ‘On the essential numerical range, the essential spectrum, and a problem of Halmos’, Acta Sci. Math. (Szeged) 33 (1972), 179192.
[6]Gohberg, I., Goldberg, S. and Kaashoek, M. A., Basic Classes of Linear Operators (Birkhäuser Verlag, Basel, 2003).
[7]Halmos, P. R., A Hilbert Space Problem Book, Graduate Texts in Mathematics, 19, 2nd edn (Springer, New York, 1982).
[8]Kato, T., Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition (Springer, Berlin, 1995).
[9]Ramesh, G., ‘Approximation methods for solving operator equations involving unbounded operators’, PhD Thesis, IIT Madras, 2008.
[10]Retherford, J. R., Hilbert Space: Compact Operators and the Trace Theorem, London Mathematical Society Student Texts 27 (Cambridge University Press, Cambridge, 1993).
[11]Rudin, W., Functional Analysis, International Series in Pure and Applied Mathematics, 2nd edn (McGraw-Hill, New York, 1991).
[12]Shkarin, S., ‘Norm attaining operators and pseudospectrum’, Integral Equations Operator Theory 64 (2009), 115136.
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Keywords

MSC classification

STRUCTURE THEOREM FOR $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{AN}$ -OPERATORS

  • G. RAMESH (a1)

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