A new criterion for 𝑘-hyponormality via weak subnormality

Curto, Raúl; Lee, Sang; Lee, Woo

doi:10.1090/S0002-9939-04-07727-5

A new criterion for $k$-hyponormality via weak subnormality
HTML articles powered by AMS MathViewer

by Raúl E. Curto, Sang Hoon Lee and Woo Young Lee PDF

Proc. Amer. Math. Soc. 133 (2005), 1805-1816 Request permission

Abstract:

In this article we obtain a criterion for $k$-hyponormality via weak subnormality. Using this criterion we recapture Spitkovskii’s subnormality criterion and give a simple proof of the main result in Gu’s preprint (2001), which describes a gap between $k$-hyponormality and ($k+1$)-hyponormality for Toeplitz operators. In addition, we notice that the minimal normal extension of a subnormal operator is exactly the inductive limit of its minimal partially normal extensions.

References

Joseph Bram, Subnormal operators, Duke Math. J. 22 (1955), 75–94. MR 68129
John B. Conway, The theory of subnormal operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR 1112128, DOI 10.1090/surv/036
Carl C. Cowen, More subnormal Toeplitz operators, J. Reine Angew. Math. 367 (1986), 215–219. MR 839133, DOI 10.1515/crll.1986.367.215
Carl C. Cowen, Hyponormal and subnormal Toeplitz operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 155–167. MR 958573
Carl C. Cowen and John J. Long, Some subnormal Toeplitz operators, J. Reine Angew. Math. 351 (1984), 216–220. MR 749683
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 10.1016/j.jpaa.2018.12.012
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, Il Bong Jung, and Sang Soo Park, A characterization of $k$-hyponormality via weak subnormality, J. Math. Anal. Appl. 279 (2003), no. 2, 556–568. MR 1974045, DOI 10.1016/S0022-247X(03)00034-9
Raúl E. Curto, Sang Hoon Lee, and Woo Young Lee, Subnormality and 2-hyponormality for Toeplitz operators, Integral Equations Operator Theory 44 (2002), no. 2, 138–148. MR 1930833, DOI 10.1007/BF01217530
Raúl E. Curto and Woo Young Lee, Joint hyponormality of Toeplitz pairs, Mem. Amer. Math. Soc. 150 (2001), no. 712, x+65. MR 1810770, DOI 10.1090/memo/0712
Raúl E. Curto and Woo Young Lee, Towards a model theory for 2-hyponormal operators, Integral Equations Operator Theory 44 (2002), no. 3, 290–315. MR 1933654, DOI 10.1007/BF01212035
Raúl E. Curto and Woo Young Lee, Subnormality and $k$-hyponormality of Toeplitz operators: a brief survey and open questions, Operator theory and Banach algebras (Rabat, 1999) Theta, Bucharest, 2003, pp. 73–81. MR 2006315
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
R. G. Douglas, Vern Paulsen, and Ke Ren Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 1, 67–71. MR 955316, DOI 10.1090/S0273-0979-1989-15700-5
Peng Fan, Note on subnormal weighted shifts, Proc. Amer. Math. Soc. 103 (1988), no. 3, 801–802. MR 947661, DOI 10.1090/S0002-9939-1988-0947661-0
C. Gu, Non-subnormal $k$-hyponormal Toeplitz operators, preprint, 2001.
P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2
P. R. Halmos, Ten years in Hilbert space, Integral Equations Operator Theory 2 (1979), no. 4, 529–564. MR 555777, DOI 10.1007/BF01691076
Ji Pu Ma and Shao Jie Zhou, A necessary and sufficient condition for an operator to be subnormal, Nanjing Daxue Xuebao Shuxue Bannian Kan 2 (1985), no. 2, 258–267 (Chinese, with English summary). MR 834313
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
Ju. L. Šmul′jan, An operator Hellinger integral, Mat. Sb. (N.S.) 49 (91) (1959), 381–430 (Russian). MR 0121662
I. M. Spitkovskiĭ, A criterion for the subnormalcy of operators in Hilbert space, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 86–87 (Russian). MR 659177
J. G. Stampfli, Which weighted shifts are subnormal?, Pacific J. Math. 17 (1966), 367–379. MR 193520