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Subnormality of Bergman-like weighted shifts. (English) Zbl 1072.47027
For \(l\geq1\), a Bergman-like weighted shift \(B_+^{(l)}\) is the weighted shift with weight sequence \(\{\alpha_n^{(l)}\}_{n=0}^\infty\), where \(\alpha_n^{(l)} = \sqrt{l-\frac{1}{n+2}}\). For \(a,b,c,d\geq0\) with \(ad-bc>0\), denote by \(S(a,b,c,d)\) the weighted shift with weight sequence \(\{\alpha_n\}_{n=0}^\infty\), where \(\alpha_n=\sqrt{\frac{an+b}{cn+d}}\). It is clear that \(B_+^{(l)}=S(l,2l-1,1,2)\) for \(l\geq1\). In the paper under review, the authors, using Schur product techniques, show that the weighted shifts \(S(a,b,c,d)\) are subnormal and, for \(p\geq1\), all \(p\)-subshifts of \(S(a,b,c,d)\) are subnormal.

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B20 Subnormal operators, hyponormal operators, etc.
Full Text: DOI
[1] Athavale, A., On joint hyponormality of operators, Proc. amer. math. soc., 103, 417-423, (1988) · Zbl 0657.47028
[2] Atkinson, K., Introduction to numerical analysis, (1989), Wiley New York
[3] Conway, J., The theory of subnormal operators, Math. surveys monogr., vol. 36, (1991), Amer. Math. Soc. Providence, RI · Zbl 0743.47012
[4] Curto, R., Joint hyponormality: A bridge between hyponormality and subnormality, Proc. sympos. pure math., 51, 69-91, (1990) · Zbl 0713.47019
[5] Curto, R., Quadratically hyponormal weighted shifts, Integral equations operator theory, 13, 49-66, (1990) · Zbl 0702.47011
[6] Curto, R., An operator-theoretic approach to truncated moment problems, (), 75-104 · Zbl 0884.47006
[7] Curto, R.; Fialkow, L., Recursively generated weighted shifts and the subnormal completion problem, Integral equations operator theory, 17, 202-246, (1993) · Zbl 0804.47028
[8] Curto, R.; Fialkow, L., Recursively generated weighted shifts and the subnormal completion problem, II, Integral equations operator theory, 18, 369-426, (1994) · Zbl 0807.47016
[9] Curto, R.; Muhly, P.; Xia, J., Hyponormal pairs of commuting operators, Oper. theory adv. appl., 35, 1-22, (1988)
[10] Curto, R.; Putinar, M., Nearly subnormal operators and moment problems, J. funct. anal., 115, 480-497, (1993) · Zbl 0817.47026
[11] R. Curto, J. Yoon, Spectral picture of 2-variable weighted shifts, in preparation · Zbl 1107.47004
[12] McCullough, S.; Paulsen, V., A note on joint hyponormality, Proc. amer. math. soc., 107, 187-195, (1989) · Zbl 0677.47018
[13] Marshall, A.W.; Olkin, I., Inequalities: the theory of majorization and its applications, (1979), Academic Press New York · Zbl 0437.26007
[14] Paulsen, V., Completely bounded maps and dilations, Pitman res. notes math. ser., vol. 146, (1986), Longman New York · Zbl 0614.47006
[15] Stampfli, J., Hyponormal operators, Pacific J. math., 12, 1453-1458, (1962) · Zbl 0129.08701
[16] Wolfram Research Inc., Mathematica, version 4.2, (2002), Wolfram Research Inc. Champaign, IL
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