• 대한수학회보
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• 제57권4호
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• pp.831-838
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• 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.

• 대한수학회지
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• 제41권1호
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• pp.1-20
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• 2004

Let A be a uniform algebra, and let $\phi$ be a self-map of the spectrum $M_A$ of A that induces a composition operator $C_{\phi}$, on A. It is shown that the image of $M_A$ under some iterate ${\phi}^n$ of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of $\phi$ converge. On the other hand, the image of the spectrum of A under $\phi$ is not hyperbolically bounded if and only if there is a subspace of $A^{**}$ "almost" isometric to ${\ell}_{\infty}$ on which ${C_{\phi}}^{**}$ "almost" an isometry. A corollary of these characterizations is that if $C_{\phi}$ is weakly compact, and if the spectrum of A is connected, then $\phi$ has a unique fixed point, to which the iterates of $\phi$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].

• Korean Journal of Mathematics
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• 제27권3호
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• pp.735-741
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• 2019

Let ${\mathcal{H}}$ be a complex Hilbert space and ${\mathcal{B}}({\mathcal{H}})$ the algebra of all bounded linear operators on ${\mathcal{H}}$. In this paper, we prove that if ${\varphi}:{\mathcal{B}}({\mathcal{H}}){\rightarrow}{\mathcal{B}}({\mathcal{H}})$ is a unital surjective bounded linear map, which preserves m- isometries m = 1, 2 in both directions, then there are unitary operators $U,V{\in}{\mathcal{B}}({\mathcal{H}})$ such that ${\varphi}(T)=UTV$ or ${\varphi}(T)=UT^{tr}V$ for all $T{\in}{\mathcal{B}}({\mathcal{H}})$, where $T^{tr}$ is the transpose of T with respect to an arbitrary but fixed orthonormal basis of ${\mathcal{H}}$.

• 대한수학회보
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• 제32권1호
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• pp.85-91
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• 1995

For a locally compact Hausdorff space, we denote by $C_0(X)$ the Banach space of all continuous complex valued functions defined on X which vanish at infinity, equipped with the usual sup norm. In case X is compact, we write C(X) instead of $C_0(X)$. A well-known Banach-Stone theorem states that the existence of an isometry between the function spaces $C_0(X)$ and $C_0(Y)$ implies X and Y are homemorphic. D. Amir [1] and M. Cambern [2] independently generalized this theorem by proving that if $C_0(X)$ and $C_0(Y)$ are isomorphic under an isomorphism T satisfying $\left\$\mid$ T \right\$\mid$ \left\$\mid$ T^1 \right\$\mid$ < 2$, then X and Y must also be homeomorphic.