Commuting Tuples of Normal Operators in Hilbert Spaces
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Complex Analysis and Operator Theory
Commuting Tuples of Normal Operators in Hilbert Spaces Hamadi Baklouti1 · Kais Feki1 Received: 4 May 2019 / Accepted: 3 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we aim to study the tensor product and the tensor sum of two jointlynormal operators. Mainly, an alternative proof is given for the result of Ch¯o and Takaguchi (Pac J Math 95(1):27–35, 1981) asserting that: if T is jointly-normal, then r (T) = T = ω(T), where r (T), ω(T) and T denote respectively the joint spectral radius, the joint numerical radius and the joint norm of an operator tuple T. It seems that this new method allows to handle more general situations, namely the operators acting on semi-hilbertian spaces. Keywords Normal operator · Tensor product · Spectral radius · Numerical radius · Operator norm Mathematics Subject Classification Primary 47B15 · 47A80 · 47A12; Secondary 46C05 · 47A05
1 Introduction Throughout this paper, H will be a complex Hilbert space, with the inner product · | · and the norm · . B(H) stands for the Banach algebra of all bounded linear operators on H. In all that follows, by an operator we mean a bounded linear operator.
Communicated by Victor Vinnikov.
B
Kais Feki [email protected] Hamadi Baklouti [email protected]
1
Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, B.P. 1171, 3000 Sfax, Tunisia 0123456789().: V,-vol
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H. Baklouti , K. Feki
Given a d-tuple T = (T1 , . . . , Td ) of operators on H, the joint norm of T is defined as T := sup
⎧ d ⎨ ⎩
k=1
1/2 Tk x2
⎫ ⎬ ; x ∈ H, x = 1 . ⎭
d : H −→ ⊕i=1 = (T1 x, . . . , Td x). It is the norm of the operator A H defined by Ax Further, the joint numerical radius of T is given by ⎧ ⎫ 1/2 d ⎨ ⎬ ω(T) = sup |Tk x | x|2 ; x ∈ H, x = 1 . ⎩ ⎭ k=1
Notice that the above two quantities was defined in [11, Definition 2]. We shall say that an operator tuple T = (T1 , . . . , Td ) is commuting if the operators Tk are pairwise commuting i.e. Ti T j = T j Ti for all i, j ∈ {1, . . . , d}. In the literature, several definitions of the spectrum of an operator tuple T = (T1 , . . . , Td ) has been done. These include Taylor spectrum σT (T), the Harte spectrum σ H (T), the joint approximate point spectrum σπ (T) and other joint spectra. Note that σT (T) is only defined for a commuting operator tuple T. The reader may consult for example [18,33] and the references therein. Associated with a d-tuple of operators T = (T1 , . . . , Td ) ∈ B(H)d , several different definitions of spectral radii exist. For more details we invite the reader to consult [6,30,31]. In 1984, the notion of the algebraic joint spectral radius of a d-tuple T was first introduced by Bunce in [9]. His definition, in the case of commuting operator tuples, reads as follows: 1 2n 1 2n n! ∗ α α n! ∗ α α T T = lim T T r (T) := inf∗ , (1.1) n→∞ n∈N α! α! d |α|=n,α∈N |α|=n, d α∈N
d Here for α = (α1 , . . . , αd ) ∈ Nd , we used α! := α1 ! · · · αd !, |α| := j=1 |α j |,
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