Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00084-9 ORIGINAL PAPER
Strongly irreducible factorization of quaternionic operators and Riesz decomposition theorem Santhosh Kumar Pamula1 Received: 15 November 2019 / Accepted: 21 July 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract Let H be a right quaternionic Hilbert space and let T be a bounded quaternionic normal operator on H . In this article, we show that T can be factorized in a strongly irreducible sense, that is, for any 𝛿 > 0 there exist a compact operator K with the norm ‖K‖ < 𝛿 , a partial isometry W and a strongly irreducible operator S on H such that
T = (W + K)S. We illustrate our result with an example. In addition, we discuss the quaternionic version of the Riesz decomposition theorem and obtain a consequence that if the S-spectrum of a bounded (need not be normal) quaternionic operator is disconnected by a pair of disjoint axially symmetric closed subsets, then the operator is strongly reducible. Keywords Axially symmetric set · Quaternionic Hilbert space · S-specturm · Strongly irreducible operator · Riesz decomposition theorem Mathematics Subject Classification 47A68 · 47A15 · 47B99
1 Introduction According to the Frobenius theorem for real division algebras, the real algebra of Hamilton quaternions is the only finite-dimensional associative division algebra that contains ℝ and ℂ as proper real subalgebras [7]. The theory of matrices over the real algebra of quaternions has been well developed in the literature parallel to the Communicated by Jan Stochel. * Santhosh Kumar Pamula [email protected] 1
Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, 8th Mile, Mysore Road, Bengaluru 560 059, India Vol.:(0123456789)
S. K. Pamula
case of real and complex matrices (see [1, 8, 22, 23] and references therein). For example, some of the fundamental results such as Schur’s canonical form and Jordan canonical form are extended to matrices with quaternion entries [23]. In particular, the Jordan canonical form infers that every square matrix over quaternions can be reduced under similarity to a direct sum of Jordan blocks. Equivalently, the Jordan canonical form determines its complete similarity invariants and establishes the structure of matrices over quaternions. Another significant aspect in this direction is the diagonalization, through which every matrix can be factorized in terms of its restriction to eigenspaces. As it is well known that every quaternionic normal matrix is diagonalizable, it is worth mentioning the result due to Weigmann [22] that a quaternion matrix is normal if and only if its adjoint is given by a polynomial of the given matrix with real coefficients. In this article, we concentrate on the class of normal operators on the right quaternionic Hilbert spaces and their factorization in a strongly irreducible sense. For this, we adopt the notion of strong irreducibility proposed by Gilfeather [10] and Jiang [14] to the class of quaternionic operators in order to replac
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