• Title/Summary/Keyword: self-adjoint operator

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BOUNDED AND UNBOUNDED OPERATORS SIMILAR TO THEIR ADJOINTS

  • Dehimi, Souheyb;Mortad, Mohammed Hichem
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.215-223
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    • 2017
  • In this paper, we establish results about operators similar to their adjoints. This is carried out in the setting of bounded and also unbounded operators on a Hilbert space. Among the results, we prove that an unbounded closed operator similar to its adjoint, via a cramped unitary operator, is self-adjoint. The proof of this result works also as a new proof of the celebrated result by Berberian on the same problem in the bounded case. Other results on similarity of hyponormal unbounded operators and their self-adjointness are also given, generalizing well known results by Sheth and Williams.

NEW EXTENSION FOR REVERSE OF THE OPERATOR CHOI-DAVIS-JENSEN INEQUALITY

  • Baharak Moosavi;Mohsen Shah Hosseini
    • Honam Mathematical Journal
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    • v.45 no.1
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    • pp.123-129
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    • 2023
  • In this paper, we introduce the reverse of the operator Davis-Choi-Jensen's inequality. Our results are employed to establish a new bound for the Furuta inequality. More precisely, we prove that, if $A,\;B{\in}{\mathcal{B}}({\mathcal{H}})$ are self-adjoint operators with the spectra contained in the interval [m, M] with m < M and A ≤ B, then for any $r{\geq}{\frac{1}{t}}>1,\,t{\in}(0,\,1)$ $A^r{\leq}({\frac{M1_{\mathcal{H}}-A}{M-m}}m^{rt}+{\frac{A-m1_{\mathcal{H}}}{M-m}}M^{rt}){^{\frac{1}{t}}}{\leq}K(m,\;M,\;r)B^r,$ where K (m, M, r) is the generalized Kantorovich constant.

A STUDY ON GENERALIZED QUASI-CLASS A OPERATORS

  • Kim, Geon-Ho;Jeon, In-Ho
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.155-159
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    • 2009
  • In this paper, we consider the operator T satisfying $T^{*k}({\mid}T^2{\mid}-{\mid}T{\mid}^2)T^k{\geq}0$ and prove that if the operator is injective and has the real spectrum, then it is self-adjoint.

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ABSTRACT RANDOM LINEAR OPERATORS ON PROBABILISTIC UNITARY SPACES

  • Tran, Xuan Quy;Dang, Hung Thang;Nguyen, Thinh
    • Journal of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.347-362
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    • 2016
  • In this paper, we are concerned with abstract random linear operators on probabilistic unitary spaces which are a generalization of generalized random linear operators on a Hilbert space defined in [25]. The representation theorem for abstract random bounded linear operators and some results on the adjoint of abstract random linear operators are given.

ADDITIVE MAPPINGS ON OPERATOR ALGEBRAS PRESERVING SQUARE ABSOLUTE VALUES

  • TAGHAVI, A.
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.51-57
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    • 2001
  • Let $\mathcal{B}(H)$ and $\mathcal{B}(K)$ denote the algebras of all bounded linear operators on Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. We show that if ${\phi}:\mathcal{B}(H){\rightarrow}\mathcal{B}(K)$ is an additive mapping satisfying ${\phi}({\mid}A{\mid}^2)={\mid}{\phi}(A){\mid}^2$ for every $A{\in}\mathcal{B}(H)$, then there exists a mapping ${\psi}$ defined by ${\psi}(A)={\phi}(I){\phi}(A)$, ${\forall}A{\in}\mathcal{B}(H)$ such that ${\psi}$ is the sum of $two^*$-homomorphisms one of which C-linear and the othere C-antilinear. We will also study some conditions implying the injective and rank-preserving of ${\psi}$.

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REMARK ON A SEGAL-LANGEVIN TYPE STOCHASTIC DIFFERENTIAL EQUATION ON INVARIANT NUCLEAR SPACE OF A Γ-OPERATOR

  • Chae, Hong Chul
    • Korean Journal of Mathematics
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    • v.8 no.2
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    • pp.163-172
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    • 2000
  • Let $\mathcal{S}^{\prime}(\mathbb{R})$ be the dual of the Schwartz spaces $\mathcal{S}(\mathbb{R})$), A be a self-adjoint operator in $L^2(\mathbb{R})$ and ${\Gamma}(A)^*$ be the adjoint operator of ${\Gamma}(A)$ which is the second quantization operator of A. It is proven that under a suitable condition on A there exists a nuclear subspace $\mathcal{S}$ of a fundamental space $\mathcal{S}_A$ of Hida's type on $\mathcal{S}^{\prime}(\mathbb{R})$) such that ${\Gamma}(A)\mathcal{S}{\subset}\mathcal{S}$ and $e^{-t{\Gamma}(A)}\mathcal{S}{\subset}\mathcal{S}$, which enables us to show that a stochastic differential equation: $$dX(t)=dW(t)-{\Gamma}(A)^*X(t)dt$$, arising from the central limit theorem for spatially extended neurons has an unique solution on the dual space $\mathcal{S}^{\prime}$ of $\mathcal{S}$.

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SELF-ADJOINT INTERPOLATION ON Ax = y IN CSL-ALGEBRA ALGL

  • Kang, Joo-Ho;Jo, Young-Soo
    • Journal of applied mathematics & informatics
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    • v.15 no.1_2
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    • pp.503-510
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    • 2004
  • Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;=\;y_i,\;for\;i\;=\;1,\;2,\;\cdots,\;n$. In this paper the following is proved: Let H be a Hilbert space and L be a commutative subspace lattice on H. Let H and y be vectors in H. Let $M_x\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_ix\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;and\;M_y\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_iy\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}. Then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y, Af = 0 for all f in ${\overline{M_x}}^{\bot}$, AE = EA for all $E\;{\in}\;L\;and\;A^{*}\;=\;A$. (2) $sup\;\{\frac{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;<\;{\infty},\;{\overline{M_u}}\;{\subset}{\overline{M_x}}$ and < Ex, y >=< Ey, x > for all E in L.

SELF-ADJOINT INTERPOLATION PROBLEMS IN ALGL

  • Kang, Joo-Ho;Jo, Young-Soo
    • Journal of applied mathematics & informatics
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    • v.14 no.1_2
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    • pp.387-395
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    • 2004
  • Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_{i}\;=\;Y_{i}$, for i = 1,2,...,n. In this article, we showed the following: Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that range(X) is dense in H. Then the following statements are equivalent: (1) There exists an operator A in AlgL such that AX = Y, $A^{*}$ = A and every E in L reduces A. (2) sup ${\frac{$\mid$$\mid${\sum_{i=1}}^n\;E_iYf_i$\mid$$\mid$}{$\mid$$\mid${\sum_{i=1}}^n\;E_iXf_i$\mid$$\mid$}$:n{\epsilon}N,f_i{\epsilon}H\;and\;E_i{\epsilon}L}\;<\;{\infty}$ and = for all E in L and all f, g in H.

SELF-ADJOINT INTERPOLATION ON Ax = y IN ALG$\cal{L}$

  • Kwak, Sung-Kon;Kang, Joo-Ho
    • Journal of applied mathematics & informatics
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    • v.29 no.3_4
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    • pp.981-986
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    • 2011
  • Given vectors x and y in a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equations $Tx_i=y_i$, for i = 1, 2, ${\cdots}$, n. In this paper the following is proved : Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and y be vectors in $\cal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE=EP_x$ for each $E{\in}\cal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\cal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and $A=A^*$. (2) sup $sup\;\{\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}\;:\;E\;{\in}\;{\cal{L}}\}$ < ${\infty}$, $y\;{\in}\;sp(x)$ and < x, y >=< y, x >.

GENERALIZED JENSEN'S EQUATIONS IN A HILBERT MODULE

  • An, Jong Su;Lee, Jung Rye;Park, Choonkil
    • Korean Journal of Mathematics
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    • v.15 no.2
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    • pp.135-148
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    • 2007
  • We prove the stability of generalized Jensen's equations in a Hilbert module over a unital $C^*$-algebra. This is applied to show the stability of a projection, a unitary operator, a self-adjoint operator, a normal operator, and an invertible operator in a Hilbert module over a unital $C^*$-algebra.

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