• 대한수학회논문집
• /
• 제9권4호
• /
• pp.847-852
• /
• 1994

Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ be the algebra of all bounded linear operator on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $I_H$ and is closed in the ultraweak operator topology on $L(H)$.

• 대한수학회보
• /
• 제30권1호
• /
• pp.65-70
• /
• 1993

In this paper we will extend some notions of bounded linear operators to some unbounded linear operators. Let H be a complex separable Hilbert space and let B(H) denote the algebra of bounded linear operators. A closed densely defind linear operator S in H, with domain domS, is called subnormal if there is a Hilbert space K containing H and a normal operator N in K(i.e., $N^{*}$N=N $N^*/)such that domS .subeq. domN and Sf=Nf for f .mem. domS. we will show that the Radjavi and Rosenthal theorem holds for some unbounded subnormal operators; if $S_{1}$ and $S_{2}$ are unbounded subnormal operators on H with dom $S_{1}$= dom $S^{*}$$_{1}$ and dom $S_{2}$=dom $S^{*}$$_{2}$ and A .mem. B(H) is injective, has dense range and $S_{1}$A .coneq. A $S^{*}$$_{2}$, then $S_{1}$ and $S_{2}$ are normal and $S_{1}$.iden. $S^{*}$$_{2}$.2}$.X>.

• 대한수학회논문집
• /
• 제37권2호
• /
• pp.409-414
• /
• 2022

Let A ∈ B(X) be a spectral operator on a non-archimedean Banach space over an algebraically closed field. In this note, we give a necessary and sufficient condition on the resolvent of A so that the discrete semigroup consisting of powers of A is uniformly-bounded.

• 대한수학회논문집
• /
• 제37권1호
• /
• pp.91-103
• /
• 2022

In this paper, we introduce Toeplitz operators and Hankel operators with complex Borel measures on the closed unit disk. When a positive measure 𝜇 on (-1, 1) is a Carleson measure, it is known that the corresponding Hankel matrix is bounded and vice versa. We show that for a positive measure 𝜇 on 𝔻, 𝜇 is a Carleson measure if and only if the Toeplitz operator with symbol 𝜇 is a densely defined bounded linear operator. We also study Hankel operators of Hilbert-Schmidt class.

• 대한수학회지
• /
• 제42권5호
• /
• pp.893-912
• /
• 2005

We discuss a certain geometric property $X_{{\theta},{\gamma}}$ of dual algebras generated by a polynomially bounded operator and property ($\mathbb{A}_{N_0,N_0}$; these are central to the study of $N_{0}\timesN_{0}$-systems of simultaneous equations of weak$^{*}$-continuous linear functionals on a dual algebra. In particular, we prove that if T $\in$ $\mathbb{A}$$^{M}$ satisfies a certain sequential property, then T $\in$ $\mathbb{A}^{M}_{N_0}(H) if and only if the algebra $A_{T}$ has property $X_{0, 1/M}$, which is an improvement of Li-Pearcy theorem in [8].

• 대한수학회보
• /
• 제50권3호
• /
• pp.1021-1027
• /
• 2013

We consider right and left Fredholm operator matrices of the form $\[\array{A&C\\T&S}\]$, which are linear and bounded on the Banach space $Z=X{\oplus}Y$.

• 대한수학회보
• /
• 제20권2호
• /
• pp.87-93
• /
• 1983

Let X and Y be normed linear spaces. An operator T defined on X with the range in Y is continuous in the sense that if a sequence {x$_{n}$} in X converges to x for the weak topology .sigma.(X.X') then {Tx$_{n}$} converges to Tx for the norm topology in Y. We shall denote the class of such operators by WC(X, Y). For example, if T is a compact operator then T.mem.WC(X, Y). In this note we discuss relationships between WC(X, Y) and the class of weakly of bounded linear operators B(X, Y). In the last section, we will consider some characters for an operator in WC(X, Y).).

• 대한수학회보
• /
• 제57권4호
• /
• pp.831-838
• /
• 2020

Let H be a complex Hilbert space and T : H → H be a bounded linear operator. Then T is said to be norm attaining if there exists a unit vector x₀ ∈ H such that ║Tx₀║ = ║T║. If for any closed subspace M of H, the restriction T|M : M → H of T to M is norm attaining, then T is called an absolutely norm attaining operator or 𝓐𝓝-operator. In this note, we discuss linear maps on B(H), which preserve the class of absolutely norm attaining operators on H.

• 호남수학학술지
• /
• 제23권1호
• /
• pp.51-57
• /
• 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 ${\psi}$.

• 대한수학회보
• /
• 제29권2호
• /
• pp.215-220
• /
• 1992

Using a matrix method, pp. Antosik and C. Swartz have obtained a series of nice properties of bounded multiplier convergent (BMC) series on metric linear spaces ([1],[8],[9]). In this paper, we establish a basic property of BMC series on topological vector spaces which is a generalization of a result due to J. Batt([2], Th.2). From this, we have obtained a kind of inclusion theorem of operator spaces. This theorem yields a nice result on infinite systems of linear equations.