Essential Norm Of Composition Operator Research Articles

Essential norm of composition operators on Harmonic Bloch spaces

In this paper we characterize essential norm of composition operators on the spaces of Harmonic Bloch functions. We obtain some upper and lower bounds for the essential norm of composition operators between Harmonic Bloch function spaces. In some cases we find exact value of the essential norm such as in Theorems 2.7 and 2.11. These results extends the similar results that were proven for composition operators on Bloch spaces.

Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space $A_{\alpha}^{\psi}$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $\psi$ satisfies the so-called $\Delta^{2}$--condition. In addition we prove that this condition characterizes those $A_{\alpha}^{\psi}$ on which every composition operator is bounded or order bounded into the Orlicz space $L_{\alpha}^{\psi}$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $\psi$ satisfies the $\Delta^{2}$--condition, a composition operator is compact on $A_{\alpha}^{\psi}$ if and only if it is order bounded into the so-called Morse-Transue space $M_{\alpha}^{\psi}$. Our results stand in the unit ball of $\mathbb{C}^{N}$.

Topological Structure of the Spaces of Composition Operators on Hilbert Spaces of Dirichlet Series

In this paper we study some topological properties of the space of bounded composition operators on some Hilbert spaces of Dirichlet series. We fi rst obtain formulas for the norms and essential norms of composition operators and di fferences of composition operators on Hilbert spaces of Dirichlet series. Then we give a characterization of the isolated points in the topological space of bounded composition operators on some Hilbert spaces of Dirichlet series. Finally we obtain sufficient conditions such that two composition operators are in the same path component. We show, among other results, that all compact composition operators are in the same path component. For a certain class of frequencies we give complete description of path components.

The associated weight and the essential norm of weighted composition operators

For an almost radial and typical weight $v$ and any weight $w$, we characterize the continuity, compactness and we estimate the essential norm of weighted composition operators $u C_{\varphi}$, acting from the weighted Banach spaces of analytic functions $H_{v}^{\infty}$ into $H_{w}^{\infty}$, in terms of the quotients of the $w$-norm of the product of $u$ with $\varphi^n$ and the $v$-norm of the $n$th power of the identity function on $\Bbb D$, where $u: \mathbb{D} \to \mathbb{C}$ and $\varphi: \mathbb{D} \to \mathbb{D}$ are analytic. As a consequence, we estimate the essential norm of composition operators $C_\varphi$ (in terms of $\varphi^n$) acting on $\mu$-Bloch spaces, for very general weights $\mu$. We also characterize continuity and compactness of weighted composition operators $uC_\varphi$ acting on $\log$-Bloch space.

Erratum and addendum to the paper "Compactness and essential norms of composition operators on Orlicz-Lorentz spaces" [Applied Mathematics Letters 25 (2012) 1778--1783

Erratum and addendum to the paper "Compactness and essential norms of composition operators on Orlicz-Lorentz spaces" [Applied Mathematics Letters 25 (2012) 1778--1783

An upper bound of the essential norm of composition operators between weighted bergman spaces

In this paper, we define the generalized counting functions in the higher dimensional case and give an upper bound of the essential norms of composition operators between the weighted Bergman spaces on the unit ball in terms of these counting functions. The sufficient condition for such operators to be bounded or compact is also given.

Composition Operators on Cesàro Function Spaces

The compact, invertible, Fredholm, and closed range composition operators are characterized. We also make an effort to compute the essential norm of composition operators on the Cesàro function spaces.

A note on the essential norm of composition operators from H p (B N ) to H q (B N )

The authors give an upper bound of the essential norms of composition operators between Hardy spaces of the unit ball in terms of the counting function in the higher dimensional value distribution theory defined by Professor S. S. Chern. The sufficient condition for such operators to be bounded or compact is also given.

Essential norms of composition operators between Bloch type spaces in the polydisk

In 2010, Ruhan Zhao obtained the essential norms of composition operators between Bloch type spaces in the disk by the nth power of the induced analytic function. This paper will generalize Zhao’s results to the polydisk. Unlike the case of the composition operators on the unit disk, the essential norms are different for the cases \({p \in (0,1)}\) and p ≥ 1.

Weighted Banach spaces of harmonic functions

We study Banach spaces of harmonic functions on open sets of \(\mathbb R ^{N}\) or \(\mathbb C ^N\) endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.

Essential norms of composition operators between weighted bergman spaces of the unit disc

In this paper, we express the essential norms of composition operators between weighted Bergman spaces of the unit disc in terms of the generalized Nevanlinna counting function.

An upper bound of essential norm of composition operator on H 2(B n )

The authors give an upper bound of the essential norm of a composition operator on H2(Bn), which involves the counting function in the higher dimensional value distribution theory defined by S. S. Chern. A criterion is also given to assure that the composition operator on H2(Bn) is bounded or compact.

Essential norm of composition operators on the Hardy space H 1 and the weighted Bergman spaces $${A_{\alpha}^{p}}$$ on the ball

We estimate the essential norm of a composition operator acting on the Hardy space H1 and the weighted Bergman spaces \({A_{\alpha}^{p}}\) on the unit ball. In passing, we recover (and somehow simplify the proof of) parts of the recent article by Demazeux, dealing with the same question for H1 of the unit disc. We also estimate the essential norm of a composition operator acting on \({A_{\alpha}^{p}}\) in terms of the angular derivatives of \({\phi}\), under a mild condition on \({\phi}\).

Compactness and essential norms of composition operators on Orlicz–Lorentz spaces

In this work, we present necessary and sufficient conditions for compactness of the composition operator on Orlicz–Lorentz spaces and determine upper and lower estimates for the essential norm of the composition operator on these spaces.

The essential norm of a composition operator on the minimal Möbius invariant space

We derive a formula for the essential norm of a composition operator on the minimal Mobius invariant space of analytic functions. This extends a recent result due to Wulan and Xiong, and completes the picture of the situation in the Besov space setting. Our methods carry over to the case of the Bergman space A1, so we are able to complement a result of Vukoti¢ concerning the essential norm of an operator on that space. Moreover, we show that the essential norm of a non-compact composition operator is at least 1. We also obtain lower bounds depending on the behavior of the symbol near the boundary, and calculate the order of magnitude of the essential norm of composition operators induced by finite Blaschke products.

The Essential Norm of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

Norms and essential norms of composition operators from H ∞ to general weighted Bloch spaces in the polydisk

Let U n be the unit polydisk of C n and a holomorphic self-map of U n . H ∞ (U n ) and B α log (U n )denote the space of bounded holomorphic functions and the space of general weighted Bloch functions defined on U n , respectively, where a>0. This paper gives some estimates of the norm and essential norm of the composition operator Cinduced by from H ∞ (U n )t oB αg (U n ). As applications, some characterizations of the boundedness and compactness of Cfrom H ∞ (U n )t oB αg (U n )are obtained. Moreover, we also characterize the weak compactness of the composition operator C. MSC(2000): Primary 47B33; Secondary 32A37

Essential norms of composition operators between Hardy spaces of the unit disc

The authors express the essential norms of composition operators between Hardy spaces of the unit disc in terms of the natural Nevanlinna counting function.

Essential Norm of Composition Operators on Banach Spaces of Hölder Functions

Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

Essential norm of composition operators between weighted Hardy spaces

An asymptotic formula for the essential norm of composition operators acting between two weighted Hardy spaces H w 1 and H w 2 , where w 1 and w 2 are two admissible weight functions, is given. The boundedness of the operators is also characterized.