Quasi-similarity of weak contractions

Author:
Pei Yuan Wu

Journal:
Proc. Amer. Math. Soc. **69** (1978), 277-282

MSC:
Primary 47A45

MathSciNet review:
0482302

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 part and part. As a corollary, two c.n.u. weak contractions are quasi-similar to each other if and only if their parts and parts are quasi-similar to each other, respectively. We also completely determine when c.n.u. weak contractions and contractions are quasi-similar to normal operators.

**[1]**Constantin Apostol,*Operators quasisimilar to a normal operator*, Proc. Amer. Math. Soc.**53**(1975), no. 1, 104–106. MR**0402522**, 10.1090/S0002-9939-1975-0402522-1**[2]**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**0394251****[3]**Ion Colojoară and Ciprian Foiaş,*Theory of generalized spectral operators*, Gordon and Breach, Science Publishers, New York-London-Paris, 1968. Mathematics and its Applications, Vol. 9. MR**0394282****[4]**John B. Conway and Pei Yuan Wu,*The splitting of 𝐴(𝑇₁⊕𝑇₂) and related questions*, Indiana Univ. Math. J.**26**(1977), no. 1, 41–56. MR**0425673****[5]**R. G. Douglas,*On the operator equation 𝑆*𝑋𝑇=𝑋 and related topics*, Acta Sci. Math. (Szeged)**30**(1969), 19–32. MR**0250106****[6]**Ali A. Jafarian,*Weak contractions of Sz.-Nagy and Foiaş are decomposable*, Rev. Roumaine Math. Pures Appl.**22**(1977), no. 4, 489–497. MR**0448149****[7]**T. L. Kriete III,*Similarity of canonical models*, Bull. Amer. Math. Soc.**76**(1970), 326–330. MR**0254650**, 10.1090/S0002-9904-1970-12462-4**[8]**C. R. Putnam,*An inequality for the area of hyponormal spectra*, Math. Z.**116**(1970), 323–330. MR**0270193****[9]**Béla Sz.-Nagy and Ciprian Foiaș,*Harmonic analysis of operators on Hilbert space*, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR**0275190****[10]**Pei Yuan Wu,*On nonorthogonal decompositions of certain contractions*, Acta Sci. Math. (Szeged)**37**(1975), no. 3-4, 301–306. MR**0399897**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47A45

Retrieve articles in all journals with MSC: 47A45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0482302-4

Keywords:
Completely nonunitary weak contractions,
quasi-similarity,
parts,
parts,
normal operators

Article copyright:
© Copyright 1978
American Mathematical Society