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Spectral properties of $p$-hyponormal and log-hyponormal operators
Bhagwati P. Duggal,전인호 대한수학회 2005 대한수학회보 Vol.42 No.3
In this paper it is proved that for p-hyponormal orlog-hyponormal operator A there exist an associated hyponormaloperator T, a quasi-anity X and an injection operator Y suchthat TX = XA and AY = Y T. The operator A and T have thesame spectral picture. We apply these results to give brief proofsof some well known spectral properties ofp-hyponormal and log-hyponormal operators, amongst them that the spectrum is a con-tinuous function on these classes of operators.
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.
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.
Harmonic Extension on an LB-Fractal
Bhagwati Prasad,Kunti Mishra 보안공학연구지원센터 2015 International Journal of Hybrid Information Techno Vol.8 No.9
The intent of the paper is to establish the energy relationship on adjacent graphs of LB-fractal constructed iteratively through an iterated function system containing five contraction maps. The harmonic matrices on the LB-fractals are obtained through harmonic extension. The normal derivatives are also found for the same.
OPERATORS A, B FOR WHICH THE ALUTHGE TRANSFORM AB IS A GENERALISED n-PROJECTION
Bhagwati P. Duggal,김인현 대한수학회 2023 대한수학회보 Vol.60 No.6
A Hilbert space operator $A\in\B$ is a generalised \linebreak $n$-projection, denoted $A\in (G-n-P)$, if ${A^*}^n=A$. $(G-n-P)$-operators $A$ are normal operators with finitely countable spectra $\sigma(A)$, subsets of the set $\{0\}\cup\{\sqrt[n+1]{1}\}$. The Aluthge transform $\a$ of $A\in\B$ may be $(G-n-P)$ without $A$ being $(G-n-P)$. For doubly commuting operators $A, B\in\B$ such that $\sigma(AB)=\sigma(A)\sigma(B)$ and $\|A\|\|B\|\leq \left\|\c\right\|$, $\c\in (G-n-P)$ if and only if $A=\left\|\a\right\|(A_{00}\oplus(A_{0}\oplus A_u))$ and $B=\left\|\b\right\|(B_0\oplus B_u)$, where $A_{00}$ and $B_0$, and $A_0\oplus A_u$ and $B_u$, doubly commute, $A_{00}B_0$ and $A_0$ are 2 nilpotent, $A_u$ and $B_u$ are unitaries, $A^{*n}_u=A_u$ and $B^{*n}_u=B_u$. Furthermore, a necessary and sufficient condition for the operators $\alpha A$, $\beta B$, $\alpha \a$ and $\beta \b$, $\alpha=\frac{1}{\left\|\a\right\|}$ and $\beta=\frac{1}{\left\|\b\right\|}$, to be $(G-n-P)$ is that $A$ and $B$ are spectrally normaloid at $0$.
k-TH ROOTS OF p-HYPONORMAL OPERATORS
DUGGAL BHAGWATI P.,JEON IN Ho,KO AND EUNGIL Korean Mathematical Society 2005 대한수학회보 Vol.42 No.3
In this paper we prove that if T is a k-th root of a phyponormal operator when T is compact or T$^{n}$ is normal for some integer n > k, then T is (generalized) scalar, and that if T is a k-th root of a semi-hyponormal operator and have the property $\sigma$(T) is contained in an angle < 2$\pi$/k with vertex in the origin, then T is subscalar.