• Title, Summary, Keyword: Fredholm operator

Search Result 33, Processing Time 0.037 seconds

PUSHCHINO DYNAMICS OF INTERNAL LAYER

  • Yum, Sang Sup
    • Korean Journal of Mathematics
    • /
    • v.12 no.1
    • /
    • 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).

  • PDF

Generalized Weyl's Theorem for Some Classes of Operators

  • Mecheri, Salah
    • Kyungpook Mathematical Journal
    • /
    • v.46 no.4
    • /
    • 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.

  • PDF

PROPERTIES OF kth-ORDER (SLANT TOEPLITZ + SLANT HANKEL) OPERATORS ON H2(𝕋)

  • Gupta, Anuradha;Gupta, Bhawna
    • Communications of the Korean Mathematical Society
    • /
    • v.35 no.3
    • /
    • 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 kth-order slant Toeplitz operator with symbol 𝜙 and $S^k_{H_{\xi}}$ is a kth-order slant Hankel operator with symbol ξ. The spectral properties of operators Sk𝓜(𝜙,𝜙) (or simply Sk𝓜(𝜙)) are investigated on H2(𝕋). More precisely, it is proved that for k = 2, the Coburn's type theorem holds for Sk𝓜(𝜙). The conditions under which operators Sk𝓜(𝜙) commute are also explored.

A NOTE ON WEIGHTED COMPOSITION OPERATORS ON MEASURABLE FUNCTION SPACES

  • Jbbarzadeh, M.R.
    • Journal of the Korean Mathematical Society
    • /
    • v.41 no.1
    • /
    • pp.95-105
    • /
    • 2004
  • In this paper we will consider the weighted composition operators W = $uC_{\tau}$ between $L^{p}$$(X,\sum,\mu$) spaces and Orlicz spaces $L^{\phi}$$(X,\sum,\mu$) generated by measurable and non-singular transformations $\tau$ from X into itself and measurable functions u on X. We characterize the functions u and transformations $\tau$ that induce weighted composition operators between $L^{p}$ -spaces by using some properties of conditional expectation operator, pair (u,${\gamma}$) and the measure space $(X,\sum,\mu$). Also, some other properties of these types of operators will be investigated.

SOME PROPERTIES OF INVARIANT SUBSPACES IN BANACH SPACES OF ANALYTIC FUNCTIONS

  • Hedayatian, K.;Robati, B. Khani
    • Honam Mathematical Journal
    • /
    • v.29 no.4
    • /
    • pp.523-533
    • /
    • 2007
  • Let $\cal{B}$ be a reflexive Banach space of functions analytic on the open unit disc and M be an invariant subspace of the multiplication operator by the independent variable, $M_z$. Suppose that $\varphi\;\in\;\cal{H}^{\infty}$ and $M_{\varphi}$ : M ${\rightarrow}$ M, defined by $M_{\varphi}f={\varphi}f$, is the operator of multiplication by ${\varphi}$. We would like to investigate the spectrum and the essential spectrum of $M_{\varphi}$ and we are looking for the necessary and sufficient conditions for $M_{\varphi}$ to be a Fredholm operator. Also we give a sufficient condition for a sequence $\{w_n\}$ to be an interpolating sequence for $\cal{B}$. At last the commutant of $M_{\varphi}$ under certain conditions on M and ${\varphi}$ is determined.

GENERALIZED BROWDER, WEYL SPECTRA AND THE POLAROID PROPERTY UNDER COMPACT PERTURBATIONS

  • Duggal, Bhaggy P.;Kim, In Hyoun
    • Journal of the Korean Mathematical Society
    • /
    • v.54 no.1
    • /
    • pp.281-302
    • /
    • 2017
  • For a Banach space operator $A{\in}B(\mathcal{X})$, let ${\sigma}(A)$, ${\sigma}_a(A)$, ${\sigma}_w(A)$ and ${\sigma}_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator A is polaroid (resp., left polaroid), if the points $iso{\sigma}(A)$ (resp., $iso{\sigma}_a(A)$) are poles (resp., left poles) of the resolvent of A. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A{\in}B(\mathcal{X})$, we prove that a sufficient condition for: (i) A+K to have SVEP on the complement of ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) for every compact operator $K{\in}B(\mathcal{X})$ is that ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) has no holes; (ii) A + K to be polaroid (resp., left polaroid) for every compact operator $K{\in}B(\mathcal{X})$ is that iso${\sigma}_w(A)$ = ∅ (resp., $iso{\sigma}_{aw}(A)$ = ∅). It is seen that these conditions are also necessary in the case in which the Banach space $\mathcal{X}$ is a Hilbert space.

The essential point spectrum of a regular operator

  • Lee, Woo-Young;Lee, Hong-Youl;Han, Young-Min
    • Bulletin of the Korean Mathematical Society
    • /
    • v.29 no.2
    • /
    • pp.295-300
    • /
    • 1992
  • In [5] it was shown that if T .mem. L(X) is regular on a Banach space X, with finite dimensional intersection T$^{-1}$ (0).cap.T(X) and if S .mem. L(X) is invertible, commute with T and has sufficiently small norm then T - S in upper semi-Fredholm, and hence essentially one-one, in the sense that the null space of T - S is finite dimensional ([4] Theorem 2; [5] Theorem 2). In this note we extend this result to incomplete normed space.

  • PDF

kth-ORDER ESSENTIALLY SLANT WEIGHTED TOEPLITZ OPERATORS

  • Gupta, Anuradha;Singh, Shivam Kumar
    • Communications of the Korean Mathematical Society
    • /
    • v.34 no.4
    • /
    • pp.1229-1243
    • /
    • 2019
  • The notion of $k^{th}$-order essentially slant weighted Toeplitz operator on the weighted Lebesgue space $L^2({\beta})$ is introduced and its algebraic properties are investigated. In addition, the compression of $k^{th}$-order essentially slant weighted Toeplitz operators on the weighted Hardy space $H^2({\beta})$ is also studied.

CONTINUITY OF APPROXIMATE POINT SPECTRUM ON THE ALGEBRA B(X)

  • Sanchez-Perales, Salvador;Cruz-Barriguete, Victor A.
    • Communications of the Korean Mathematical Society
    • /
    • v.28 no.3
    • /
    • pp.487-500
    • /
    • 2013
  • In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${\lambda}{\in}{\sigma}_{s-F}(T){\setminus}\overline{{\rho}^{\pm}_{s-F}(T)}$ and ${\in}$ > 0, the ball $B({\lambda},{\in})$ contains a component of ${\sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.