Yoenha Kim

OPERATORS WITH THE SINGLE VALUED EXTENSION PROPERTY

In this Paper We Study some Operators With the single valued extension property. In particular, we investigate the Helton class of an operator and an triangular operator matrix T.

Characterizations of binormal composition operators with linear fractional symbols on H2

For an analytic function ? : D ? D , the composition operator C? is the operator on the Hardy space H2 defined by C?f = f???? for all f in H2. In this paper, we give necessary and sufficient conditions for the composition operator C? to be binormal where the symbol ? is a linear fractional selfmap of D . Furthermore, we show that C? is binormal if and only if it is centered when ? is an automorphism of D or ?(z) = sz + t, |s| + |t| ? 1. We also characterize several properties of binormal composition operators with linear fractional symbols on H2.

On the helton class of p-hyponormal operators

In this paper we show that the Helton class of p-hyponormal operators has scalar extensions. As a corollary we get that each operator in the Helton class of p-hyponormal operators has a nontrivial invariant subspace if its spectrum has its interior in the plane.

INHERITED PROPERTIES THROUGH THE HELTON CLASS OF AN OPERATOR

In this paper we show that Helton class preserves the nilpotent and finite ascent properties. Also, we show some relations on non-transitivity and decomposability between operators and their Helton classes. Finally, we give some applications in the Helton class of weighted shifts.

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.

SOME REMARKS ON THE HELTON CLASS OF AN OPERATOR

In this paper we study some properties of the Helton class of an operator. In particular, we show that the Helton class preserves the quasinilpotent property and Dunfords boundedness condition (B). As corollaries, we get that the Helton class of some quadratically hyponor- mal operators or decomposable subnormal operators satisfles Dunfords boundedness condition (B).

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.