arXiv subjects
Publications and source records attributed to Uğur Gül.
In this paper we prove that the Cesaro operator $\mathcal{C}$ in $\ell^{2}$, the Hilbert space of square summable sequences, is essentially normal, i.e. the commutator $[\mathcal{C}^{\ast},\mathcal{C}]:=\mathcal{C}^{\ast}\mathcal{C}-\mathcal{C}\mathcal{C}^{\ast}$ is a compact operator on $\ell^{2}$.
In this paper we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application we prove that for "quasi-parabolic" composition operators the spectra and the essential spectra are equal.
In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_α^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $φ(z)=$ $z+ψ(z)$, where $ψ\in$ $H^{\infty}(\mathbb{H})$ and $\Im(ψ(z)) > ε> 0$. We especially examine the case where $ψ$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.
Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $Γ$ is partially ordered. Let $Γ^{+}\subsetΓ$ be the semigroup of positive elements in $Γ$. The Hardy space $H^{2}(G)$ is the closed subspace of $L^{2}(G)$ consisting of functions whose Fourier transforms are supported on $Γ^{+}$. In this paper we consider the C*-algebra $C^{*}(\mathcal{T}(G)\cup F(C(\dot{Γ^{+}})))$ generated by Toeplitz operators with continuous symbols on $G$ which vanish at infinity and Fourier multipliers with symbols which are continuous on one point compactification of $Γ^{+}$ on the Hilbert-Hardy space $H^{2}(G)$. We characterize the character space of this C*-algebra using a theorem of Power.
In this paper we consider the C*-algebra $C^{*}(\{C_φ\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.
This work is a generalization of the results in [Gul] to bi-disc case. As in [Gul], quasi-parabolic composition operators on the Hilbert-Hardy space of the bi-disc are written as a linear combination of Toeplitz operators and Fourier multipliers. The C*-algebra generated by Toeplitz operators and Fourier multipliers on the Hilbert-Hardy space of the bi-disc is written as the tensor product of the similar C*-algebra in one variable with itself. As a result we find a nontrivial set lying inside the essential spectra of quasi-parabolic composition operators.