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November 2017 Supporting vectors of continuous linear operators
Clemente Cobos-Sánchez, Francisco Javier García-Pacheco, Soledad Moreno-Pulido, Sol Sáez-Martínez
Ann. Funct. Anal. 8(4): 520-530 (November 2017). DOI: 10.1215/20088752-2017-0016

Abstract

The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between the supporting vectors of a real matrix A and the Tikhonov regularization min xRnAxb+αx reaching the conclusion that, by an appropriate choice of b and α, the supporting vectors of A can be obtained via solving the Tikhonov regularization min xRnAxb+αx.

Citation

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Clemente Cobos-Sánchez. Francisco Javier García-Pacheco. Soledad Moreno-Pulido. Sol Sáez-Martínez. "Supporting vectors of continuous linear operators." Ann. Funct. Anal. 8 (4) 520 - 530, November 2017. https://doi.org/10.1215/20088752-2017-0016

Information

Received: 4 October 2016; Accepted: 8 January 2017; Published: November 2017
First available in Project Euclid: 29 June 2017

zbMATH: 1383.65055
MathSciNet: MR3717174
Digital Object Identifier: 10.1215/20088752-2017-0016

Subjects:
Primary: 15A18‎
Secondary: 15A60, 47L25

Rights: Copyright © 2017 Tusi Mathematical Research Group

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Vol.8 • No. 4 • November 2017
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