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APPLICATIONS OF HILBERT SPACE DISSIPATIVE NORM
Kubrusly, Carlos S.,Levan, Nhan Korean Mathematical Society 2012 대한수학회보 Vol.49 No.1
The concept of Hilbert space dissipative norm was introduced in [8] to obtain necessary and sufficient conditions for exponential stability of contraction semigroups. In the present paper we show that the same concept can also be used to derive further properties of contraction semigroups, as well as to characterize strongly stable semigroups that are not exponentially stable.
BIISOMETRIC OPERATORS AND BIORTHOGONAL SEQUENCES
Kubrusly, Carlos,Levan, Nhan Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3
It is shown that a pair of Hilbert space operators V and W such that $V^*W=I$ (called a biisometric pair) shares some common properties with unilateral shifts when orthonormal bases are replaced with biorthogonal sequences, and it is also shown how such a pair of biisometric operators yields a pair of biorthogonal sequences which are shifted by them. These are applied to a class of Laguerre operators on $L^2[0,{\infty})$.
Biisometric operators and biorthogonal sequences
Carlos Kubrusly,Nhan Levan 대한수학회 2019 대한수학회보 Vol.56 No.3
It is shown that a pair of Hilbert space operators $V$ and $W$ such that ${V^*W=I}$ (called a biisometric pair) shares some common properties with unilateral shifts when orthonormal bases are replaced with biorthogonal sequences, and it is also shown how such a pair of biisometric operators yields a pair of biorthogonal sequences which are shifted by them. These are applied to a class of Laguerre operators on $L^2[0,\infty)$.
Carlos Kubrusly,Nhan Levan Korean Mathematical Society 2023 대한수학회보 Vol.60 No.3
Erratum/Addendum to the paper "Biisometric operators and biorthogonal sequences" [Bull. Korean Math. Soc. 56 (2019), No. 3, pp. 585-596].
APPLICATIONS OF HILBERT SPACE DISSIPATIVE NORM
Carlos S. Kubrusly,Nhan Levan 대한수학회 2012 대한수학회보 Vol.49 No.1
The concept of Hilbert space dissipative norm was introduced in [8] to obtain necessary and sufficient conditions for exponential stabil-ity of contraction semigroups. In the present paper we show that the same concept can also be used to derive further properties of contraction semigroups, as well as to characterize strongly stable semigroups that are not exponentially stable.
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>
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.
WEYL@S THEOREMS FOR POSINORMAL OPERATORS
DUGGAL BHAGWATI PRASHAD,KUBRUSLY CARLOS Korean Mathematical Society 2005 대한수학회지 Vol.42 No.3
An operator T belonging to the algebra B(H) of bounded linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator $P{\in}B(H)$ such that $TT^*\;=\;T^*PT$. A posinormal operator T is said to be conditionally totally posinormal (resp., totally posinormal), shortened to $T{\in}CTP(resp.,\;T{\in}TP)$, if to each complex number, $\lambda$ there corresponds a positive operator $P_\lambda$ such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P_{\lambda}^{\frac{1}{2}}(T-{\lambda}I)|^{2}$ (resp., if there exists a positive operator P such that $|(T-{\lambda}I)^{\ast}|^{2}\;=\;|P^{\frac{1}{2}}(T-{\lambda}I)|^{2}\;for\;all\;\lambda)$. This paper proves Weyl's theorem type results for TP and CTP operators. If $A\;{\in}\;TP$, if $B^*\;{\in}\;CTP$ is isoloid and if $d_{AB}\;{\in}\;B(B(H))$ denotes either of the elementary operators $\delta_{AB}(X)\;=\;AX\;-\;XB\;and\;\Delta_{AB}(X)\;=\;AXB\;-\;X$, then it is proved that $d_{AB}$ satisfies Weyl's theorem and $d^{\ast}_{AB}\;satisfies\;\alpha-Weyl's$ theorem.
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)
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.