Positive definiteness and commutativity of operators

Stochel, Jan

doi:10.1090/S0002-9939-98-04075-1

Positive definiteness and commutativity of operators
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Proc. Amer. Math. Soc. 126 (1998), 431-440 Request permission

Abstract:

It is shown that an $n$–tuple of bounded linear operators on a complex Hilbert space, which is positive definite in the sense of Halmos, must be commutative. Some generalizations of this result to the case of pairs of unbounded operators are obtained.

References

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Earl A. Coddington, Formally normal operators having no normal extensions, Canadian J. Math. 17 (1965), 1030–1040. MR 200719, DOI 10.4153/CJM-1965-098-8
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
Josef Daneš, On local spectral radius, Časopis Pěst. Mat. 112 (1987), no. 2, 177–187 (English, with Russian and Czech summaries). MR 897643, DOI 10.21136/CPM.1987.118306
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
J. Górniak and A. Weron, Aronszajn-Kolmogorov type theorems for positive definite kernels in locally convex spaces, Studia Math. 69 (1980/81), no. 3, 235–246. MR 647140, DOI 10.4064/sm-69-3-235-246
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177
Palle E. T. Jorgensen, Commutative algebras of unbounded operators, J. Math. Anal. Appl. 123 (1987), no. 2, 508–527. MR 883705, DOI 10.1016/0022-247X(87)90327-1
P. Masani, Dilations as propagators of Hilbertian varieties, SIAM J. Math. Anal. 9 (1978), no. 3, 414–456. MR 500218, DOI 10.1137/0509027
W. Mlak, Conditionally positive definite functions on linear spaces, Ann. Polon. Math. 42 (1983), 187–239. MR 728082, DOI 10.4064/ap-42-1-187-239
Konrad Schmüdgen, A formally normal operator having no normal extension, Proc. Amer. Math. Soc. 95 (1985), no. 3, 503–504. MR 806097, DOI 10.1090/S0002-9939-1985-0806097-0
Jan Stochel, On the Sz.-Nagy boundedness condition on nonabelian involutory semigroups, Spectral theory of linear operators and related topics (Timişoara/Herculane, 1983) Oper. Theory Adv. Appl., vol. 14, Birkhäuser, Basel, 1984, pp. 251–260. MR 789623
J. Stochel, Dilatability of sesquilinear form-valued kernels, Ann. Polon. Math. 48 (1988), no. 1, 1–29. MR 931386, DOI 10.4064/ap-48-1-1-29
Jan Stochel, Moment functions on real algebraic sets, Ark. Mat. 30 (1992), no. 1, 133–148. MR 1171099, DOI 10.1007/BF02384866
J. Stochel and F. H. Szafraniec, On normal extensions of unbounded operators. I, J. Operator Theory 14 (1985), no. 1, 31–55. MR 789376
J. Stochel and F. H. Szafraniec, $\mathcal C^\infty$–vectors and boundedness, Ann. Polon. Math. 66 (1997), 223–238.
F. H. Szafraniec, Dilations on involution semigroups, Proc. Amer. Math. Soc. 66 (1977), no. 1, 30–32. MR 473873, DOI 10.1090/S0002-9939-1977-0473873-1
F. H. Szafraniec, Boundedness of the shift operator related to positive definite forms: an application to moment problems, Ark. Mat. 19 (1981), no. 2, 251–259. MR 650498, DOI 10.1007/BF02384482
F. H. Szafraniec, Kato-Protter type inequalities, bounded vectors and the exponential function, Ann. Polon. Math. 51 (1990), 303–312. MR 1094002, DOI 10.4064/ap-51-1-303-312
Béla Sz.-Nagy, Extensions of linear transformations in Hilbert space which extend beyond this space (Appendix to Frigyes Riesz and Béla Sz.-Nagy, “Functional analysis”), Frederick Ungar Publishing Co., New York, 1960. Translated from the French by Leo F. Boron. MR 0117561