An inequality for the area of hyponormal spectra

Putnam, C. R.

doi:10.1007/bf01111839

Cited by 92 publications

(35 citation statements)

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“…Since A,' is subnormal and o(N'x) has planar area zero (cf. [9, p. 126]), it follows from Putnam's theorem [8] that N'x is normal. This completes the proof.…”

Section: ])mentioning

confidence: 99%

Quasi-Similarity of Weak Contractions

Wu¹

1978

Proceedings of the American Mathematical Society

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Abstract. Let T be a completely nonunitary (c.n.u.) weak contraction (in the sense of Sz.-Nagy and Foias). We show that T is quasi-similar to the direct sum of its C0 part and Cu part. As a corollary, two c.n.u. weak contractions are quasi-similar to each other if and only if their C0 parts and Cu parts are quasi-similar to each other, respectively. We also completely determine when c.n.u. weak contractions and C0 contractions are quasi-similar to normal operators. 323]).Assume that T is a weak contraction which is also completely nonunitary (c.n.u.), that is, T has no nontrivial reducing subspace on which T is a unitary operator. For such a contraction, Sz.-Nagy and Foia § obtained a C0-Cxx decomposition and then found a variety of invariant subspaces which furnish its spectral decomposition (cf. [9, Chapter VIII]). In this note we are going to supplement other interesting properties of such contractions. We show that a c.n.u. weak contraction is quasi-similar to the direct sum of its C0 part and C,, part. Although the proof is not difficult, some of its interesting applications justify the elaboration here. An immediate corollary is that two such contractions are quasi-similar to each other if and only if their C0 parts are quasi-similar and their C,, parts are quasi-similar to each other. This is, in turn, used to show that two quasi-similar weak contractions have equal spectra. Another interesting consequence is that a c.n.u. weak contraction is quasi-similar to a normal operator if and only if its C0 part is. The latter can be shown to be equivalent to the condition that its minimal function is a Blaschke product with simple zeros, thus completely settling the question when a c.n.u. weak contraction is quasi-similar to a normal operator.Before we start to prove our main theorem, we provide some background work for our notations and terminology. The main reference is [9].

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“…Since A,' is subnormal and o(N'x) has planar area zero (cf. [9, p. 126]), it follows from Putnam's theorem [8] that N'x is normal. This completes the proof.…”

Section: ])mentioning

confidence: 99%

Quasi-Similarity of Weak Contractions

Wu¹

1978

Proceedings of the American Mathematical Society

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“…In view of Theorem 1 of [15], the last inclusion is impossible since T is not normal. Hence, σ(T ) = {a + ib ∈ C : ( The following example shows that H = {k(λ) : λ ∈ B a (T )} is not sufficient condition for T to have a connected spectrum, and gives a negative answer to Question B of [21] (see also [22]).…”

Section: Examples and Commentsmentioning

confidence: 99%

Bounded point evaluations for cyclic Hilbert space operators

Bourhim

2003

Appl. Gen. Topol.

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“…An alternate proof can be obtained from an inequality concerning hyponormal operators derived in [4].…”

Section: T Omentioning

confidence: 99%

“…See Theorem 1 of [4], where the additional hypothesis that T be completely IIDII 1 and so the second part of (3.2) follows from (ii) of (I). (See Sz.…”

Section: Iig(8)dg(8)ii -mentioning

confidence: 99%