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Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08
Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08
Pomona Faculty Publications and Research
A matrix T∈Mn(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above.
On The Closure Of The Complex Symmetric Operators: Compact Operators And Weighted Shifts, Stephan Ramon Garcia, Daniel E. Poore '11
On The Closure Of The Complex Symmetric Operators: Compact Operators And Weighted Shifts, Stephan Ramon Garcia, Daniel E. Poore '11
Pomona Faculty Publications and Research
We study the closure $\bar{CSO}$ of the set $CSO$ of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in $\bar{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.