Sungeun Jung

Normal and cohyponormal weighted composition operators on H 2

In this paper we study normal and cohyponormal weighted composition operators on the Hardy space H 2. We show that if W f,ϕ is cohyponormal, then f is outer and ϕ is univalent. Moreover, we prove that when the composition map ϕ has the Denjoy–Wolff point in the open unit disk, W f,ϕ is cohyponormal if and only if it is normal; in this case, f and ϕ can be expressed as linear fractional maps. As a corollary, we find the polar decomposition of the cohyponormal operator W f,ϕ . Finally, we examine the commutant of a cohyponormal weighted composition operator.

Self-commutators of invertible weighted composition operators on

In this paper, we consider the self-commutator of an invertible weighted composition operator on the Hardy space where is continuous on . We show that both the self-commutator and the anti-self-commutator are expressed as compact perturbations of Toeplitz operators. Moreover, we give an alternative proof for the result in [2] that is unitary exactly when is an automorphism of and where , is the reproducing kernel at for , and is a constant with . We next show that when for all , the weighted composition operator is normal if and only if the composition operator is unitary and is constant on . We also provide some spectral properties of and .

Iterated Aluthge transforms of composition operators on H2

In this paper, we study various properties of the iterated Aluthge transforms of the composition operators Cφ and Cσ where φ(z) = az + (1 - a) and for 0 < a < 1. We express the iterated Aluthge transforms and as weighted composition operators with linear fractional symbols. As a corollary, we prove that and are not quasinormal but binormal. In addition, we show that and are quasisimilar for all non-negative integers n and m. Finally, we show that and converge to normal operators in the strong operator topology.

Commutators of weighted composition operators

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of 𝔻 such that φ(ζ) and ψ(ζ) belong to ∂𝔻 for some ζ ∈ ∂𝔻 and u, v ∈ H∞ are continuous on ∂𝔻 with u(ζ)v(ζ) ≠ 0, then is compact on H2 if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H2. We verify that if φ is an analytic self-map of 𝔻 with Denjoy–Wolff point b ∈ 𝔻 and u ∈ H∞\{0}, then every weighted composition operator in the commutant {Wu, φ}′ has {f ∈ H2 : f(b) = 0} as its nontrivial invariant subspace.

On operators which are power similar to hyponormal operators

In this paper, we study power similarity of operators. In par ticular, we show that if T 2 PS(H ) (defined below) for some hyponormal operator H , thenT is subscalar. From this result, we obtain that such an operator with rich sp ectrum has a nontrivial invariant subspace. Moreover, we consider invariant and hyp erinvariant subspaces for T 2 PS(H ).

Composition operators for which and commute

We define . In this paper, we characterize composition operators and their adjoints which belong to , where the maps are linear fractional selfmaps of the open unit disk into itself. If is an automorphism of or , then the case for is precisely when it is normal. When , we also prove that if , then either or , which implies that the only binormal composition operators with and are normal. Moreover, we show that if and is not normal, then implies that and is neither real nor purely imaginary, while ensures that and is real. Finally, we study composition operators in where is an analytic selfmap into . In particular, this operator has the single-valued extension property.

On 2×2 operator matrices

In this paper, we show that some 2× 2 operator matrices have scalar extensions. In particular, we focus on some 2-hyponormal operators and their generalizations. As a corollary, we get that such operator matrices have nontrivial invariant subspaces if their spectra have nonempty interiors in the complex plane. Mathematics subject classification (2010): Primary 47A11, Secondary 47A15, 47B20.