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Journal of Operator Theory

Volume 81, Issue 1, Winter 2019  pp. 81-105.

Taylor spectra and common invariant subspaces through the Duggal and generalized Aluthge transforms for commuting $n$-tuples of operators

Authors:  Jaewoong Kim (1), Jasang Yoon (2)
Author institution: (1) Department of Mathematics, Korea Military Academy, Seoul, 01805, Korea
(2) School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas 78539, U.S.A.

Summary:  In the first part of this paper, we introduce two notions of multivariable Duggal transforms (toral and spherical), and study their basic properties including hyponormality and norm-continuity. In the second part, we study how the Taylor spectrum and Taylor essential spectrum of $2 $-variable weighted shifts behave under the toral and spherical Duggal transforms including generalized Aluthge transforms. In the last part, we investigate nontrivial common invariant subspaces between the toral (respectively spherical) Duggal transform and the original $n$-tuple of bounded operators with dense ranges. We also study the sets of common invariant subspaces among them.

Keywords: Duggal transforms, generalized Aluthge transforms, common invariant subspaces, Taylor spectra, commuting $n$-tuples, $2$-variable weighted shifts

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