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Operator equations and subscalarity

100%
Jung S., Ko E.
Studia Mathematica
|
2014
|
tom 225
|
nr 2
97-113
EN
We consider the system of operator equations ABA = A² and BAB = B². Let (A,B) be a solution to this system. We give several connections among the operators A, B, AB, and BA. We first prove that A is subscalar of finite order if and only if B is, which is equivalent to the subscalarity of AB or BA with finite order. As a corollary, if A is subscalar and its spectrum has nonempty interior, then B has a nontrivial invariant subspace. We also provide examples of subscalar operator matrices. Moreover, we deal with algebraicity, power boundedness, and quasitriangularity, using some power properties obtained from the operator equations.
2
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On class A operators

81%
Jung S., Ko E., Lee M.-J.
Studia Mathematica
|
2010
|
tom 198
|
nr 3
249-260
EN
We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.
3
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On (A,m)-expansive operators

81%
Jung S., Kim Y., Ko E., Lee J.
Studia Mathematica
|
2012
|
tom 213
|
nr 1
3-23
EN
We give several conditions for (A,m)-expansive operators to have the single-valued extension property. We also provide some spectral properties of such operators. Moreover, we prove that the A-covariance of any (A,2)-expansive operator T ∈ ℒ(ℋ ) is positive, showing that there exists a reducing subspace ℳ on which T is (A,2)-isometric. In addition, we verify that Weyl's theorem holds for an operator T ∈ ℒ(ℋ ) provided that T is (T*T,2)-expansive. We next study (A,m)-isometric operators as a special case of (A,m)-expansive operators. Finally, we prove that every operator T ∈ ℒ(ℋ ) which is (T*T,2)-isometric has a scalar extension.
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