• Communications of the Korean Mathematical Society
• /
• v.35 no.1
• /
• pp.151-159
• /
• 2020

We introduce the notion of (m, n)-paranormal operators and (m, n)^⁎-paranormal operators on Hilbert space and study their properties. We also characterize these operators. Examples of operators are given which are (m, n)-paranormal but not (m, n)^⁎-paranormal, and vice-versa.

• Korean Journal of Mathematics
• /
• v.24 no.1
• /
• pp.65-70
• /
• 2016

J. G. Stampfli proved that if a bounded linear operator T on a Hilbert space ${\mathfrak{H}}$ satisfies ($G_1$) property, then the Riesz projection $P_{\lambda}$ associated with ${\lambda}{\in}iso{\sigma}$(T) is self-adjoint and $P_{\lambda}{\mathfrak{H}}=(T-{\lambda})^{-1}(0)=(T^*-{\bar{\lambda}})^{-1}(0)$. In this note we show that Stampfli''s result is generalized to an nilpotent extension of an operator having ($G_1$) property.

• Journal of the Korean Mathematical Society
• /
• v.41 no.3
• /
• pp.423-433
• /
• 2004

In this paper we obtained the following: Let H. be a Hilbert space and (equation omitted) be a subspace lattice on H. Let X and Y be operators acting on H. If the range of X is dense in H, then the following are equivalent: (1) there exists an operator A in Alg(equation omitted) such that AX = Y, (2) sup (equation omitted) Moreover, if condition (2) holds, we may choose the operator A such that ∥A∥ = K.

• Honam Mathematical Journal
• /
• v.29 no.4
• /
• pp.569-576
• /
• 2007

Let $\cal{H}$ be a separable, infinite dimensional, complex Hilbert space. We call "an operator $\cal{T}$ acting on $\cal{H}$ has a star moment sequence supported on a set K" when there exist nonzero vectors u and v in $\cal{H}$ and a positive Borel measure ${\mu}$ such that <$T^{*j}T^ku$, v> = ${^\int\limits_{K}}\;{{\bar{z}}^j}\;{{\bar{z}}^k}\;d\mu$ for all j, $k\;\geq\;0$. We obtain a characterization to find a representing star moment measure and discuss some related properties.

• Journal of the Korean Mathematical Society
• /
• v.33 no.1
• /
• pp.205-216
• /
• 1996

In [13], Ptak and Vrbova proved that if T is a bounded normal operator T on a complex Hilbert space H, then the ranges of the spectral projections can be represented in the form $$ \varepsilon(F)H = \bigcap_{\lambda\notinF} (T - \lambda I) H for all closed subsets F of C, $$ where $\varepsilon$ denotes the spectral measure associated with T.

• Journal of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.311-319
• /
• 1995

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ be the algebra of all bounded linear operaors on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $I_H$ and is closed in the $weak^*$ topology on $L(H)$. Note that the ultraweak operator topology coincides with the $weak^*$ topology on $L(H)$ (see [5]).

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.747-753
• /
• 2006

We say operators A, B on Hilbert space satisfy Fuglede-Putnam theorem if AX = X B for some X implies $A^*X=XB^*$. We show that if either (1) A is p-hyponormal and $B^*$ is a class y operator or (2) A is a class y operator and $B^*$ is p-hyponormal, then A, B satisfy Fuglede-Putnam theorem.

• Journal of applied mathematics & informatics
• /
• v.18 no.1_2
• /
• pp.513-524
• /
• 2005

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. Let L be a subspace lattice on H. We obtained a necessary and sufficient condition for the existence of an interpolating operator A which is in AlgL.

• Korean Journal of Mathematics
• /
• v.16 no.4
• /
• pp.545-551
• /
• 2008

We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.12 no.1
• /
• pp.1-6
• /
• 2008

Duggal-Jeon-Kubrusly([2]) introduced Hilbert space operator T satisfying property ${\mid}T{\mid}^2{\leq}{\mid}T^2{\mid}$, where ${\mid}T{\mid}=(T^*T)^{1/2}$. In this paper we extend this property to general version, namely property B(n). In addition, we construct examples which distinguish the classes of operators with property B(n) for each $n{\in}\mathbb{N}$.