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Moment infinitely divisible weighted shifts. (English) Zbl 07032878
Summary: We say that a weighted shift $$W_\alpha$$ with (positive) weight sequence $$\alpha : \alpha_0, \alpha_1, \dots$$ is moment infinitely divisible (MID) if, for every $$t > 0$$, the shift with weight sequence $$\alpha ^t: \alpha_0^t, \alpha_1^t, \dots$$ is subnormal. Assume that $$W_{\alpha}$$ is a contraction, i.e., $$0 < \alpha_i \leq 1$$ for all $$i \geq 0$$. We show that such a shift $$W_\alpha$$ is MID if and only if the sequence $$\alpha$$ is log completely alternating. This enables the recapture or improvement of some previous results proved rather differently. We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Mathematica
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##### References:
 [1] Agler, J., Hypercontractions and subnormality, J. Oper. Theory, 13, 203-217, (1985) · Zbl 0593.47022 [2] Athavale, A., On completely hyperexpansive operators, Proc. Am. Math. Soc., 124, 3745-3752, (1996) · Zbl 0863.47017 [3] Athavale, A.; Ranjekar, A., Bernstein functions, complete hyperexpansivity and subnormality-I, Integral Equ. Oper. Theory, 43, 253-263, (2002) · Zbl 1052.47014 [4] Bram, J., Subnormal operators, Duke Math. J., 22, 15-94, (1955) · Zbl 0064.11603 [5] Bhatia, R., Infinitely divisible matrices, Amer. Math. Monthly, 113, 221-235, (2006) · Zbl 1132.15019 [6] Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, Berlin (1984) · Zbl 0619.43001 [7] Comtet, L.: Advanced Combinatorics. D. Reidel Publ. Co., Dordretch (1974) · Zbl 0283.05001 [8] Conway, J.B.: The Theory of Subnormal Operators. In: Mathematical Surveys and Monographs, vol. 36. American Mathematical Society, Providence, RI, pp. xvi$$+$$436 (1991). ISBN:0-8218-1536-9 [9] Cowen, C.; Long, J., Some subnormal Toeplitz operators, J. Reine Angew. Math., 351, 216-220, (1984) · Zbl 0532.47019 [10] Cui, J.; Duan, Y., Berger measure for $$S(a, b, c, d)$$, J. Math. Anal. Appl., 413, 202-211, (2014) · Zbl 1331.47031 [11] Curto, R., Quadratically hyponormal weighted shifts, Integral Equ. Oper. Theory, 13, 49-66, (1990) · Zbl 0702.47011 [12] Curto, R.; Exner, GR, Berger measure for some transformations of subnormal weighted shifts, Integral Equ. Oper. Theory, 84, 429-450, (2016) · Zbl 1341.47035 [13] Curto, R.; Park, SS, $$k$$-hyponormality of powers of weighted shifts via Schur products, Proc. Am. Math. Soc., 131, 2761-2769, (2002) · Zbl 1022.47022 [14] Curto, R.; Poon, YT; Yoon, J., Subnormality of Bergman-like weighted shifts, J. Math. Anal. Appl., 308, 334-342, (2005) · Zbl 1072.47027 [15] Exner, GR, On $$n$$-contractive and $$n$$-hypercontractive operators, Integral Equ. Oper. Theory, 58, 451-468, (2006) · Zbl 1118.47012 [16] Exner, GR, Aluthge transforms and $$n$$-contractivity of weighted shifts, J. Oper. Theory, 61, 419-438, (2009) · Zbl 1199.47146 [17] Gellar, R.; Wallen, LJ, Subnormal weighted shifts and the Halmos-Bram criterion, Proc. Japan Acad., 46, 375-378, (1970) · Zbl 0217.45501 [18] Lee, SH; Lee, WY; Yoon, J., The mean transform of bounded linear operators, J. Math. Anal. Appl., 410, 70-81, (2014) · Zbl 1327.47016 [19] Sato, K.: Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, (2004). (Reprint of 1999 translation of Kahou Katei (Japanese), Kinokuniya Press, 1990.) [20] Shields, A., Weighted shift operators and analytic function theory, Math. Surv., 13, 49-128, (1974) · Zbl 0303.47021 [21] Sholapurkar, VM; Athavale, A., Completely and alternatingly hyperexpansive operators, J. Oper. Theory, 43, 43-68, (2000) · Zbl 0992.47012 [22] Mathematica, Version 8.0, Wolfram Research Inc., Champaign, IL, (2011)
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