𝑘-hyponormality of powers of weighted shifts via Schur products

Curto, Raúl; Park, Sang

doi:10.1090/S0002-9939-02-06805-3

$k$-hyponormality of powers of weighted shifts via Schur products
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by Raúl Curto and Sang Soo Park

Proc. Amer. Math. Soc. 131 (2003), 2761-2769

DOI: https://doi.org/10.1090/S0002-9939-02-06805-3

Published electronically: November 13, 2002

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Abstract:

We characterize $k$-hyponormality and quadratic hyponormality of powers of weighted shifts using Schur product techniques.

References

Ameer Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103 (1988), no. 2, 417–423. MR 943059, DOI 10.1090/S0002-9939-1988-0943059-X
J. Bae, G. Exner and I.B. Jung, Criteria for positively quadratically hyponormal weighted shifts, Proc. Amer. Math. Soc., to appear.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
John B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR 634507
Raúl E. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13 (1990), no. 1, 49–66. MR 1025673, DOI 10.1007/BF01195292
Raúl E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, Operator theory: operator algebras and applications, Part 2 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 69–91. MR 1077422, DOI 10.1090/pspum/051.2/1077422
Raúl E. Curto and Lawrence A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math. 17 (1991), no. 4, 603–635. MR 1147276
Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations Operator Theory 17 (1993), no. 2, 202–246. MR 1233668, DOI 10.1007/BF01200218
Raúl E. Curto and Lawrence A. Fialkow, Recursively generated weighted shifts and the subnormal completion problem. II, Integral Equations Operator Theory 18 (1994), no. 4, 369–426. MR 1265443, DOI 10.1007/BF01200183
Raúl E. Curto and Il Bong Jung, Quadratically hyponormal weighted shifts with two equal weights, Integral Equations Operator Theory 37 (2000), no. 2, 208–231. MR 1769810, DOI 10.1007/BF01192423
Raúl E. Curto, Il Bong Jung, and Woo Young Lee, Extensions and extremality of recursively generated weighted shifts, Proc. Amer. Math. Soc. 130 (2002), no. 2, 565–576. MR 1862138, DOI 10.1090/S0002-9939-01-06079-8
Raúl E. Curto and Mihai Putinar, Existence of nonsubnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 373–378. MR 1091568, DOI 10.1090/S0273-0979-1991-16079-9
Raúl E. Curto and Mihai Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. 115 (1993), no. 2, 480–497. MR 1234402, DOI 10.1006/jfan.1993.1101
Raúl E. Curto, Paul S. Muhly, and Jingbo Xia, Hyponormal pairs of commuting operators, Contributions to operator theory and its applications (Mesa, AZ, 1987) Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 1–22. MR 1017663
Il Bong Jung and Chunji Li, A formula for $k$-hyponormality of backstep extensions of subnormal weighted shifts, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2343–2351. MR 1823917, DOI 10.1090/S0002-9939-00-05844-5
Il Bong Jung and Sang Soo Park, Quadratically hyponormal weighted shifts and their examples, Integral Equations Operator Theory 36 (2000), no. 4, 480–498. MR 1759824, DOI 10.1007/BF01232741
Il Bong Jung and Sang Soo Park, Cubically hyponormal weighted shifts and their examples, J. Math. Anal. Appl. 247 (2000), no. 2, 557–569. MR 1769094, DOI 10.1006/jmaa.2000.6879
Scott McCullough and Vern Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), no. 1, 187–195. MR 972236, DOI 10.1090/S0002-9939-1989-0972236-8
Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
J. G. Stampfli, Which weighted shifts are subnormal?, Pacific J. Math. 17 (1966), 367–379. MR 193520
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
Evgenii M. Vorob′ev, Symmetry analysis of nonlinear PDE with a “Mathematica” program SYMMAN, J. Nonlinear Math. Phys. 3 (1996), no. 1-2, 85–89. Symmetry in nonlinear mathematical physics, Vol. 2 (Kiev, 1995). MR 1374119, DOI 10.2991/jnmp.1996.3.1-2.8