Some new classes of complex symmetric operators

By Stephan Ramon Garcia, W. Wogen. 2009

205 citations recorded by the source corpus.

We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7606c6a8823177eb8371af3b8a2cd86d47b225b4

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In Applied Mathematics

1. Stephan Ramon Garcia
2. William T. Ross
3. J. Lee
4. Sungeun Jung
5. W. Wogen
6. Eungil Ko
7. M. Lee
8. Yoenha Kim

In Mathematical Physics

1. Stephan Ramon Garcia
2. William T. Ross
3. J. Lee
4. W. Wogen
5. Sungeun Jung
6. Eungil Ko
7. M. Lee
8. Yoenha Kim

Citations in the corpus

• Complex symmetric weighted composition operators on H2(D) (2014, background): “Thus Cφ is complex symmetric from [18].”
• Recent Progress on Truncated Toeplitz Operators (2011, background): “In particular, Theorem 12 says each truncated Toeplitz operator is a complex symmetric operator, a class of Hilbert space operators which has undergone much recent study [24, 35, 45–50, 54, 55, 61, 63, 71, 72, 74, 103–107].”
• Complex symmetric partial isometries (2009, methodology): “The aim of this note is to complete the classification of complex symmetric partial isometries which was started in [10].”
• On local spectral properties of complex symmetric operators (2011, background): “This class includes all normal operators, Hankel operators, compressed Toeplitz operators, and the Volterra integration operator, and there are a lot of consequences and applications concerning complex symmetric operators (see [10–16,26], etc.”