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On the range kernel orthogonality of derivations
Duggal, B.P. TOPOLOGY AND GEOMETRY RESEARCH CENTER 1998 Proceedings of the Topology and Geometry Research Vol.9 No.-
Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of operators on H into itself. The generalized derivation δA,B : B(H) → B(H) is defined by δA,B (X) = AX - XB; let △A,B : B(H) → B(H) be defined by △A,B (X) = AXB - X, and let d A,B denote δA,B or △A,B . Given that S ∈ Cp (the Schatten p-class, 1 < p < ∞) and that the pair of operators A and B* satisfies a Putnam-Fuglede commutativity theorem type property, we consider here a necessary and sufficient condition for ∥dA,B (X) + S∥p ≥ ∥S∥ p to hold for all X ∈ C p.
REMARKS ON SPECTRAL PROPERTIES OF p-HYPONORMAL AND LOG-HYPONORMAL OPERATORS
DUGGAL BHAGWATI P.,JEON, IN-HO Korean Mathematical Society 2005 대한수학회보 Vol.42 No.3
In this paper it is proved that for p-hyponormal or log-hyponormal operator A there exist an associated hyponormal operator T, a quasi-affinity X and an injection operator Y such that TX = XA and AY = YT. The operator A and T have the same spectral picture. We apply these results to give brief proofs of some well known spectral properties of p-hyponormal and loghyponormal operators, amongst them that the spectrum is a continuous function on these classes of operators.
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.
On *-paranormal contractions and properties for *-class A operators
Duggal, B.P.,Jeon, I.H.,Kim, I.H. North Holland [etc.] 2012 Linear algebra and its applications Vol.436 No.5
An operator T@?B(H) is called a *-class A operator if |T<SUP>2</SUP>|≥|T<SUP>*</SUP>|<SUP>2</SUP>, and T is said to be *-paranormal if @?T<SUP>*</SUP>x@?<SUP>2</SUP>≤@?T<SUP>2</SUP>x@? for every unit vector x in H. In this paper we show that *-paranormal contractions are the direct sum of a unitary and a C<SUB>.0</SUB> completely non-unitary contraction. Also, we consider the tensor products of *-class A operators.
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)
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>
Duggal, B.P.,Jeon, I.H.,Kim, I.H. Elsevier 2012 Linear algebra and its applications Vol.436 No.9
<P><B>Abstract</B></P><P>Let QA denote the class of bounded linear Hilbert space operators <I>T</I> which satisfy the operator inequality <SUP>T∗</SUP>|<SUP>T2</SUP>|T⩾<SUP>T∗</SUP>|T<SUP>|2</SUP>T. It is proved that if T∈QA is a contraction, then either <I>T</I> has a nontrivial invariant subspace or <I>T</I> is a proper contraction and the nonnegative operator D=<SUP>T∗</SUP>(|<SUP>T2</SUP>|-|T<SUP>|2</SUP>)T is strongly stable. It is shown that if T∈QA is a contraction with Hilbert–Schmidt defect operator such that <SUP>T-1</SUP>(0)⊆<SUP><SUP>T∗</SUP>-1</SUP>(0), then <I>T</I> is completely non–normal if and only if T∈<SUB>C10</SUB>, and a commutativity theorem is proved for contractions T∈QA. Let <SUB>Tu</SUB> and <SUB>Tc</SUB> denote the unitary part and the cnu part of a contraction <I>T</I>, respectively. We prove that if A=<SUB>Au</SUB>⊕<SUB>Ac</SUB> and B=<SUB>Bu</SUB>⊕<SUB>Bc</SUB> are QA-contractions such that <SUB>μ<SUB>Ac</SUB></SUB><∞, then <I>A</I> and <I>B</I> are quasi-similar if and only <SUB>Au</SUB> and <SUB>Bu</SUB> are unitarily equivalent and <SUB>Ac</SUB> and <SUB>Bc</SUB> are quasi-similar.</P>
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.
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.
ON WEYL'S THEOREM FOR QUASI-CLASS A OPERATORS
Duggal Bhagwati P.,Jeon, In-Ho,Kim, In-Hyoun Korean Mathematical Society 2006 대한수학회지 Vol.43 No.4
Let T be a bounded linear operator on a complex infinite dimensional Hilbert space $\scr{H}$. We say that T is a quasi-class A operator if $T^*\|T^2\|T{\geq}T^*\|T\|^2T$. In this paper we prove that if T is a quasi-class A operator and f is a function analytic on a neigh-borhood or the spectrum or T, then f(T) satisfies Weyl's theorem and f($T^*$) satisfies a-Weyl's theorem.