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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 $L^2[0,\infty)$.
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.
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.
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 $L^2[0,{\infty})$.
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].
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.
WEYL'S THEOREMS FOR POSINORMAL OPERATORS
Bhagwati Prashad Duggal,Carlos Kubrusly 대한수학회 2005 대한수학회지 Vol.42 No.3
An operator T belonging to the algebra B(H) of bound-ed linear transformations on a Hilbert H into itself is said to be posinormal if there exists a positive operator P 2 B(H)such that TT¤ = T¤PT. A posinormal operator T is said to be condi-tionally totally posinormal (resp., totally posinormal), shortened to T 2 CTP (resp., T 2 TP), if to each complex number ¸ there corre-sponds a positive operator P¸ such that j(T¡¸I)¤j2 = jP 1/2¸ (T¡¸I)j2(resp., if there exists a positive operator P such that j(T ¡¸I)¤j2 =jP1/2 (T ¡ ¸I)j2 for all ¸). This paper proves Weyl's theorem type results for TP and CTP operators. If A 2 TP, if B¤ 2 CTP is isoloid and if dAB 2 B(B(H)) denotes either of the elementary op-erators ±AB(X) = AX ¡XB and 4AB(X) = AXB ¡X, then it is proved that dAB satis¯esWeyl's theorem and d¤AB satis¯es a-Weyl's theorem.