Certain numerical radius contraction operators

Authors:
Takayuki Furuta and Ritsuo Nakamoto

Journal:
Proc. Amer. Math. Soc. **29** (1971), 521-524

MSC:
Primary 47.40

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285924-2

MathSciNet review:
0285924

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper an operator *T* means a bounded linear operator on a complex Hilbert space *H*. The numerical radius norm of an operator *T*, is defined by for every unit vector *x* in *H*. An operator *T* is said to be a numerical radius contraction if . We shall give some theorems on certain numerical radius contraction operators and related results in consequence of these theorems.

Our central result is that an idempotent numerical radius contraction is a projection. Finally we prove that a periodic numerical radius contraction is the direct sum of zero and a unitary operator, that is to say, normal and partial isometric.

**[1]**C. A. Berger,*A strange dilation theorem*, Notices Amer. Math. Soc.**12**(1965), 590. Abstract #625-152.**[2]**Takayuki Furuta,*On the class of paranormal operators*, Proc. Japan Acad.**43**(1967), 594–598. MR**0221302****[3]**Takayuki Furuta, Midori Horie, and Ritsuo Nakamoto,*A remark on a class of operators*, Proc. Japan Acad.**43**(1967), 607–609. MR**0221304****[4]**Takayuki Furuta and Ritsuo Nakamoto,*Some theorems on certain contraction operators*, Proc. Japan Acad.**45**(1969), 565–567. MR**0253079****[5]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[6]**V. Istrăţescu,*On some hyponormal operators*, Pacific J. Math.**22**(1967), 413–417. MR**0213893****[7]**Vasile Istrăţescu, Teishirô Saitô, and Takashi Yoshino,*On a class of operators*, Tôhoku Math. J. (2)**18**(1966), 410–413. MR**0209860**, https://doi.org/10.2748/tmj/1178243383**[8]**T. Kato,*Remarks on the numerical radius*(unpublished).**[9]**Carl Pearcy,*An elementary proof of the power inequality for the numerical radius*, Michigan Math. J.**13**(1966), 289–291. MR**0201976****[10]**J. G. Stampfli,*Normality and the numerical range of an operator*, Bull. Amer. Math. Soc.**72**(1966), 1021–1022. MR**0212599**, https://doi.org/10.1090/S0002-9904-1966-11625-7

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47.40

Retrieve articles in all journals with MSC: 47.40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0285924-2

Keywords:
Complex Hilbert space,
bounded linear operator,
numerical radius,
numerical radius contraction,
spectraloid,
normaloid,
paranormal operator,
hyponormal operator,
idempotent operator,
periodic operator,
partial isometric operator

Article copyright:
© Copyright 1971
American Mathematical Society