• 대한수학회논문집
• /
• 제11권3호
• /
• pp.657-663
• /
• 1996

In this paper we give some conditions under which the Weyl spectrum of an operator satisfies the spectral mapping theorem for analytic functions. Also we show that Weyl's theorem holds for p(T) where T is an operator of M-power class (N) and p is a polynomial on a neighborhood of $\sigam(T)$. Finally we answer an old question of Oberai.

• 대한수학회보
• /
• 제58권2호
• /
• pp.327-347
• /
• 2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators B�� is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero H-Toeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

• 호남수학학술지
• /
• 제30권2호
• /
• pp.391-397
• /
• 2008

The existence of solutions and the occurrence of a Hopf bifurcation for the free boundary problems with Pushchino dynamics was shown in [1]. In this paper, we shall show a Hopf bifurcation occurs for the free boundary for Pushchino dynamics with a spatial dependency.

• Korean Journal of Mathematics
• /
• 제12권1호
• /
• pp.7-14
• /
• 2004

The existence of solutions and the occurence of a Hopf bifurcation for the free boundary problem with Pushchino dynamics was shown in [3]. In this paper we shall show a Hopf bifurcation occurs for the free boundary which is given by (1).

• 대한수학회논문집
• /
• 제37권1호
• /
• pp.75-89
• /
• 2022

In this paper, we prove new fixed point theorems for single-valued and multi-valued weakly Picard operators in complete metric spaces and give several examples. As applications, we give several results to Fredholm integral equation.

• Kyungpook Mathematical Journal
• /
• 제46권4호
• /
• pp.553-563
• /
• 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set ${\sigma}_{B{\omega}}(A)$ of all ${\lambda}{\in}\mathbb{C}$ such that $A-{\lambda}I$ is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem ${\sigma}_{B{\omega}}(A)={\sigma}(A)$\E(A), and the B-Weyl spectrum ${\sigma}_{B{\omega}}(A)$ of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalizedWeyl's theorem holds for the case where A is an algebraically ($p,\;k$)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

• 대한수학회논문집
• /
• 제33권3호
• /
• pp.809-821
• /
• 2018

In the present paper, we extend the main results of Jeribi in [6] to weighted and pseudo-weighted spectra of operators in a nonseparable Hilbert space ${\mathcal{H}}$. We investigate the characterization, the stability and some properties of these weighted and pseudo-weighted spectra.

• Nonlinear Functional Analysis and Applications
• /
• 제28권4호
• /
• pp.989-1004
• /
• 2023

In this paper, we are motivated to evaluate and investigate the necessary conditions for the fractional Volterra Fredholm integro-differential equation involving the ς-Hilfer fractional derivative. The given problem is converted into an equivalent fixed point problem by introducing an operator whose fixed points coincide with the solutions to the problem at hand. The existence and uniqueness results for the given problem are derived by applying Krasnoselskii and Banach fixed point theorems respectively. Furthermore, we investigate the convergence of approximated solutions to the same problem using the modified Adomian decomposition method. An example is provided to illustrate our findings.

• 충청수학회지
• /
• 제25권4호
• /
• pp.655-668
• /
• 2012

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

• 대한수학회논문집
• /
• 제35권3호
• /
• pp.855-866
• /
• 2020

For two essentially bounded Lebesgue measurable functions 𝜙 and ξ on unit circle 𝕋, we attempt to study properties of operators $S^k_{\mathcal{M}({\phi},{\xi})=S^k_{T_{\phi}}+S^k_{H_{\xi}}$ on H²(𝕋) (k ≥ 2), where $S^k_{T_{\phi}}$ is a k^th-order slant Toeplitz operator with symbol 𝜙 and $S^k_{H_{\xi}}$ is a k^th-order slant Hankel operator with symbol ξ. The spectral properties of operators S^k_{𝓜(𝜙,𝜙)} (or simply S^k_𝓜(𝜙)) are investigated on H²(𝕋). More precisely, it is proved that for k = 2, the Coburn's type theorem holds for S^k_𝓜(𝜙). The conditions under which operators S^k_𝓜(𝜙) commute are also explored.