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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.
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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.
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UPPER TRIANGULAR OPERATORS WITH SVEP
Duggal, Bhagwati Prashad Korean Mathematical Society 2010 대한수학회지 Vol.47 No.2
A Banach space operator A $\in$ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A $\in$ ($\mathcal{H}\mathcal{P}$), if every part of A is polaroid. Let $X^n\;=\;\oplus^n_{t=i}X_i$, where $X_i$ are Banach spaces, and let A denote the class of upper triangular operators A = $(A_{ij})_{1{\leq}i,j{\leq}n$, $A_{ij}\;{\in}\;B(X_j,X_i)$ and $A_{ij}$ = 0 for i > j. We prove that operators A $\in$ A such that $A_{ii}$ for all $1{\leq}i{\leq}n$, and $A^*$ have the single-valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A $\in$ A such that $A_{ii}$ $\in$ ($\mathcal{H}\mathcal{P}$) for all $1{\leq}i{\leq}n$ are polaroid and have SVEP; hence they satisfy Weyl's theorem. Furthermore, A+R satisfies Browder's theorem for all upper triangular operators R, such that $\oplus^n_{i=1}R_{ii}$ is a Riesz operator, which commutes with A.
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UPPER TRIANGULAR OPERATORS WITH SVEP
Bhagwati Prashad Duggal 대한수학회 2010 대한수학회지 Vol.47 No.2
A Banach space operator A ∈ B(X) is polaroid if the isolated points of the spectrum of A are poles of the resolvent of A; A is hereditarily polaroid, A ∈ (HP), if every part of A is polaroid. Let [수식]Xi, where Xi are Banach spaces, and let A denote the class of upper triangular operators A = (Aij )1≤i,j·n, Aij ∈ B(Xj ,Xi) and Aij = 0 for i > j. We prove that operators A ∈ A such that Aii for all 1≤ i · n, and A¤ have the single–valued extension property have spectral properties remarkably close to those of Jordan operators of order n and n-normal operators. Operators A 2 A such that Aii ∈ (HP) for all 1 ≤ i · n are polaroid and have SVEP; hence they satisfy Weyl’s theorem. Furthermore, A + R satisfies Browder’s theorem for all upper triangular operators R, such that [수식] is a Riesz operator, which commutes with A.
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On Self-commutator Approximants
Duggal, Bhagwati Prashad Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.1
Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h ${\in}$ B(X) : h is hermitian}, and J(X) = {x ${\in}$ B(X) : x = $x_1$ + $ix_2$, $x_1$ and $x_2$ ${\in}$ H(X)}. Let ${\delta}_a$ ${\in}$ B(B(X)) denote the derivation ${\delta}_a$ = ax - xa. If J(X) is an algebra and ${\delta}_a^{-1}(0){\subseteq}{\delta}_{a^*}^{-1}(0)$ for some $a{\in}J(X)$, then ${\parallel}a{\parallel}{\leq}{\parallel}a-(x^*x-xx^*){\parallel}$ for all $x{\in}J(X){\cap}{\delta}_a^{-1}(0)$. The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = $C_p$, the von Neumann-Schatten p-class, are considered.
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