In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These ...characterizations are useful in checking C-normality of an operator. We illustrate these results with examples.
For some operator A ? B(H), positive integers m and k, an operator T ? B(H) is called k-quasi-(A,m)-symmetric if T*k( mP j=0 (?1)j(m j )T*m?jATj)Tk = 0, which is a generalization of the m-symmetric ...operator. In this paper, some basic structural properties of k-quasi-(A,m)-symmetric operators are established with the help of operator matrix representation. We also show that if T and Q are commuting operators, T is k-quasi-(A,m)-symmetric and Q is n-nilpotent, then T + Q is (k + n ? 1)-quasi-(A,m + 2n ? 2)-symmetric. In addition, we obtain that every power of k-quasi-(A,m)-symmetric is also k-quasi-(A,m)-symmetric. Finally, some spectral properties of k-quasi-(A,m)-symmetric are investigated.
On complex symmetric Toeplitz operators Ko, Eungil; Lee, Ji Eun
Journal of mathematical analysis and applications,
02/2016, Letnik:
434, Številka:
1
Journal Article
Recenzirano
Odprti dostop
In this paper, we give a characterization of a complex symmetric Toeplitz operator Tφ on the Hardy space H2. Moreover, if Tφ is a complex symmetric Toeplitz operator, we provide a necessary and ...sufficient condition for Tφ to be normal. Finally, we investigate these Tφ with finite symbols.
An operator T∈B(H) is complex symmetric if there exists a conjugation C on H so that CTC=T⁎. In this paper, we characterize the complex symmetric Toeplitz operator on the Hardy and Bergman space. In ...particular, we show that the Toeplitz operator induced by the Berezin transform of a complex symmetric operator on Hardy space is also complex symmetric with the same conjugation. However, we see that this is not true for Bergman space by providing some examples.
In this article we present the polar decomposition of a bounded linear operator defined on an infinite-dimensional complex Hilbert space with respect to a conjugation. As a result, the polar ...decomposition for
C
-symmetric,
C
-skew-symmetric, and
C
-normal operators are obtained. As an another consequence the Godič –Lucenko theorem is obtained. We give two applications of the polar decomposition theorem. First, we obtain the polar decomposition of bounded antilinear operators and discuss various cases. Second, we derive an inequality for the bounded linear operator based on the Hölder-McCarthy’s inequality and derive an inequality for
C
-normal operators.
Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if
. In this paper, first, we give a representation of C-normal operators on finite ...dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem.
A bounded linear operator T:H→H is said to be complex symmetric if there exists a conjugation C on H such that CT⁎C=T. In this paper, we study the numerical ranges of complex symmetric operators. We ...show that every complex symmetric operator T on H has a small compact perturbation being complex symmetric and having a closed numerical range. We also show that this holds for skew symmetric operators but fails to hold for unitary operators.
We study when a Toeplitz operator Tϕ on the Dirichlet space of the unit disk is complex symmetric with respect to a class of conjugations and find surprisingly that the case of complex symmetries of ...Toeplitz operators according to these conjugations is very few. We also show that if Tϕ is complex symmetric, then the curve ϕ|T(T) must be nowhere winding. Furthermore, the spectrum and invertibility of complex symmetric Toeplitz operators are described.
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