Quasi-similarity of weak contractions

Wu, Pei Yuan

doi:10.1090/S0002-9939-1978-0482302-4

Quasi-similarity of weak contractions
HTML articles powered by AMS MathViewer

by Pei Yuan Wu PDF

Proc. Amer. Math. Soc. 69 (1978), 277-282 Request permission

Abstract:

Let T be a completely nonunitary (c.n.u.) weak contraction (in the sense of Sz.-Nagy and Foiaş). We show that T is quasi-similar to the direct sum of its ${C_0}$ part and ${C_{11}}$ part. As a corollary, two c.n.u. weak contractions are quasi-similar to each other if and only if their ${C_0}$ parts and ${C_{11}}$ parts are quasi-similar to each other, respectively. We also completely determine when c.n.u. weak contractions and ${C_0}$ contractions are quasi-similar to normal operators.

References

Constantin Apostol, Operators quasisimilar to a normal operator, Proc. Amer. Math. Soc. 53 (1975), no. 1, 104–106. MR 402522, DOI 10.1090/S0002-9939-1975-0402522-1
H. Bercovici, C. Foiaş, and B. Sz.-Nagy, Compléments à l’étude des opérateurs de classe $C_{O}$. III, Acta Sci. Math. (Szeged) 37 (1975), no. 3-4, 313–322 (French). MR 394251
Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Mathematics and its Applications, Vol. 9, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0394282
John B. Conway and Pei Yuan Wu, The splitting of $A(T_{1}\oplus T_{2})$ and related questions, Indiana Univ. Math. J. 26 (1977), no. 1, 41–56. MR 425673, DOI 10.1512/iumj.1977.26.26002
R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
Ali A. Jafarian, Weak contractions of Sz.-Nagy and Foiaş are decomposable, Rev. Roumaine Math. Pures Appl. 22 (1977), no. 4, 489–497. MR 448149
T. L. Kriete III, Similarity of canonical models, Bull. Amer. Math. Soc. 76 (1970), 326–330. MR 254650, DOI 10.1090/S0002-9904-1970-12462-4
C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330. MR 270193, DOI 10.1007/BF01111839
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Pei Yuan Wu, On nonorthogonal decompositions of certain contractions, Acta Sci. Math. (Szeged) 37 (1975), no. 3-4, 301–306. MR 399897