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Duggal, B.P.,Kubrusly, C.S.,Kim, I.H. Elsevier 2015 Journal of mathematical analysis and applications Vol.427 No.1
<P><B>Abstract</B></P> <P>Given a Hilbert space operator A ∈ B ( H ) with polar decomposition A = U | A | , the class A ( s , t ) , 0 < s , t ≤ 1 , consists of operators A ∈ B ( H ) such that <SUP> | <SUP> A ⁎ </SUP> | 2 t </SUP> ≤ <SUP> ( <SUP> | <SUP> A ⁎ </SUP> | t </SUP> <SUP> | A | 2 s </SUP> <SUP> | <SUP> A ⁎ </SUP> | t </SUP> ) t t + s </SUP> . Every class A ( s , t ) operator is paranormal; prominent amongst the subclasses of A ( s , t ) operators are the class A ( 1 2 , 1 2 ) consisting of w-hyponormal operators and the class A ( 1 , 1 ) consisting of (semi-quasihyponormal [16, p. 93], or) class A operators. Our aim here is threefold. We prove that A ( s , t ) operators satisfy: (i) Bishop's property (<I>β</I>), thereby providing a proof of [6, Theorem 3.1], and (ii) a Putnam–Fuglede commutativity theorem, thereby answering a question posed in [18, Conjecture 2.4]; we prove also an extension of [3, Theorem 3.4] to prove that (iii) if an A ( s , t ) operator is weakly supercyclic then it is a scalar multiple of a unitary operator.</P>
Contractions of Class Q and Invariant Subspaces
B. P. Duggal,C. S. Kubrusly,N. Levan 대한수학회 2005 대한수학회보 Vol.42 No.1
A Hilbert Space operator T is of class Q ifT^{2*}T^2-2kern1ptT^*T+I is nonnegative. Every paranormaloperator is of class Q, but class-Q operators are notnecessarily normaloid. It is shown that if a class-Qcontraction T has no nontrivial invariant subspace, then it is aproper contraction. Moreover, the nonnegative operatorQ=T^{2*}T^2-2kern1ptT^*T+I also is a propercontraction.
PARANORMAL CONTRACTIONS AND INVARIANT SUBSPACES
Duggal, B.P.,Kubrusly, C.S.,Levan, N. Korean Mathematical Society 2003 대한수학회지 Vol.40 No.6
It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T/sup 2*/T/sup 2/ - 2T/sup */T + I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contraction. Furthermore, if one of Q or D is compact, then so is the other, and Q and D are strict ontraction.
CONTRACTIONS OF CLASS Q AND INVARIANT SUBSPACES
DUGGAL, B.P.,KUBRUSLY, C.S.,LEVAN, N. Korean Mathematical Society 2005 대한수학회보 Vol.42 No.1
A Hilbert Space operator T is of class Q if $T^2{\ast}T^2-2T{\ast}T + I$ is nonnegative. Every paranormal operator is of class Q, but class-Q operators are not necessarily normaloid. It is shown that if a class-Q contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = $T^2{\ast}T^2-2T{\ast}T + I$ also is a proper contraction.
ERRATUM TO "PARANORMAL CONTRACTIONS AND INVARIANT SUBSPACES"
Duggal, B.P.,Kubrusly, C.S.,Levan, N. Korean Mathematical Society 2004 대한수학회지 Vol.41 No.4
In our paper "Paranormal contractions and invariant subspaces" published in Journal of the Korean Mathematical Society, Volume 40 (2003), Number 6, pp.933-942, the statement to observation (1) on page 935 should read:(omitted)
Paranormal contractions and invariant subspaces
B. P. Duggal,C. S. Kubrusly,N. Levan 대한수학회 2003 대한수학회지 Vol.40 No.6
It is shown that if a paranormal contraction T has no nontrivialinvariant subspace, then it is a proper contraction. Moreover, thenonnegative operatorQ=T^{2*}T^2-2kern.5ptT^*Tkern-.5pt+kern-.5ptI also is aproper contraction. If a quasihyponormal contraction has nonontrivial invariant subspace then, in addition, its defectoperator D is a proper contraction and its itself-commutator isa trace-class strict contraction. Furthermore, if one of Q orD is compact, then so is the other, and Q and D are strictcontraction.