In this note, we show some basic results related to the spectrum of a weakly selfadjoint operator

Nakamura Noboru

application/pdfIn this note, we show some basic results related to the spectrum of a weakly selfadjoint operator.ArticleToyama mathematical journal，vol.32，Page 1-4departmental bulletin pape

Nakamura, Noboru

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Toyama Math. J.Vol. 32(2009), 1-4Weakly selfadjoint operators IINoboru NakamuraAbstract. In this note, we show some basic results related to the spectrumof a weakly selfadjoint operator.1. Throughout this note, an operator means a bounded linear operator on a Hilbertspace H. An operator A is positive (denoted by A ≥ 0) if (Ax, x) ≥ 0 for allx ∈ H. For an operator A, the numerical range W (A) means the following set inthe complex plane:W (A) = {(Ax, x) ; x ∈ H, ‖ x ‖= 1}.For the set W (A) the following general facts are well-known (cf. [3, § 2.5]).(i) W (A) is convex.(ii) W (A), the closure of W (A), contains the spectrum σ(A) of A, so thatcoσ(A) ⊂W (A),where coσ(A) is the convex hull of σ(A).We define the spectral radius r(A) of an operator A byr(A) = sup{|λ|; λ ∈ σ(A)}.2000 Mathematics Subject Classification. 47A10, 47A12, 47A30, 47A63.Key words and phrases. weakly positive operator, spectrum, spectral radius, numerical range,weakly selfadjoint operator.12 Noboru NakamuraNow if σ(A) lies in the real line R and if we putmA = inf{λ ;λ ∈ σ(A)} and MA = sup{λ ;λ ∈ σ(A)},then from convexity of W (A) and the fact mA,MA ∈ σ(A), we can see that(1.1) coσ(A) = [mA,MA] =W (A) =W (A).In [5], S. Izumino and the author defined a weakly selfadjoint operator, anal-ogously as a weakly positive operator defined in [4], and gave some fundamentalresults on the operator in the light of the recent knowledge. In succession of theconsideration, in this note, we shall show some further facts related to the spectrumof the operator.2. R. Bouldin [1] (cf. [6, Theorem 2]) showed the following:Theorem B. If one of operators A and B is positive, then coσ(AB) ⊂ W (A) ·W (B).Applying this result, we show the following fact.Proposition 1. If T = AB is a weakly selfadjoint operator as the product of twoselfadjoint operators A and B, one of whose factors is positive, then(2.1) coσ(T ) ⊂ coσ(A) coσ(B).Proof. From the above (1.1) we seecoσ(A) =W (A) =W (A) and coσ(B) =W (B) =W (B).Hence we haveW (A) ·W (B) = coσ(A) coσ(B),so that by Theorem B we have the desired (2.1).From Proposition 1, we have the following:Corollary 2 (cf. [5, Corollary 2]). If T is a weakly selfadjoint operator, then(2.2) σ(T ) ⊂ R = (−∞,∞).Weakly selfadjoint operators II 3Related to Corollary 2 the following interesting fact is known between the spec-tral radius r(T ) and the norm of Re T = (T + T ∗)/2.Proposition 3 (cf. [2, Lemma 2]). If σ(T ) is contained in R, then(2.3) r(T ) ≤‖ Re T ‖.Hence, for a weakly selfadjoint T , (2.3) holds.For completeness we give a proof of (2.3): Recall thatcoσ(T ) ⊂W (T ).Using the relation Re W (T ) =W (Re T ), we haveRe W (T ) =W (Re T ) ⊂ [− ‖ Re T ‖, ‖ Re T ‖].Henceσ(T ) = Re σ(T ) ⊂ [− ‖ Re T ‖, ‖ Re T ‖],so that we obtain the desired inequality (2.3).Example. Let T = AB with A =[2 11 1]and B =[1 00 −1]. Then by easycalculation we obtain r(T ) = 1+√52 = 1.618 . . . and ‖ Re T ‖= 2, which shows theabove inequality (2.3).Acknowledgement. The author would like to thank the late Professor M. Naka-mura and Professor S. Izumino for valuable suggestions.References[1] R. Bouldin, The numerical range of a product, J. Math. Anal. Appl., 32(1970), 459-467.[2] J. I. Fujii, M. Fujii and R. Nakamoto, Norm inequalities related toMcIntosh type inequality, Nihonkai Math. J., 3 (1992), 67-72.[3] T. Furuta, Invitation to Linear Operators, From Matrices to Bounded Lin-ear Operators on a Hilbert Space, Taylor and Francis Inc., New York, (2001).4 Noboru Nakamura[4] S. Izumino and M. Nakamura, Wigner’s weakly positive operators, Sci-entiae Mathematicae Japonicae, 65 (2007), 1095-1102.[5] S. Izumino and N. Nakamura, Weakly selfadjoint operators, Math. J.Toyama Univ., 30 (2007), 45-56.[6] R. Nakamoto, On a theorem of Bouldin, Proc. Japan Acad., 47 (1971),979-981.[7] E. P. Wigner, On weakly positive operators, Canadian J. Math., 15 (1965),313-317.Noboru NakamuraToyama National College of Technology, Hongo-Machi 13, Toyama, 930-8630, JapanE-mail address: n-nakamu@nc-toyama.ac.jp(Received July 2, 2007)

Weakly selfadjoint operators II

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Weakly selfadjoint operators II

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