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- Complex symmetric operator (5)
- Hilbert space (2)
- Partial isometry (2)
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- Binormal operator (1)
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- Compact operator (1)
- Conjugation (1)
- Eigenvalue (1)
- Euclid (1)
- Foguel operator (1)
- Hilbert space operators (1)
- Hilbert’s axioms (1)
- Hyperbolic geometry (1)
- Hyperbolic metric (1)
- Idempotent (1)
- Isometry (1)
- J-self-adjoint differential operators (1)
- Kakutani shift (1)
- Model space (1)
- Nilpotent operator (1)
- Nilpotent operators of order two (1)
- Norm closure (1)
- Normal operator (1)
- Palindrome (1)
- Poincaré model (1)
- Pseudo-hyperbolic metric (1)
- Reproducing kernel (1)
- Self-similarity (1)
- Shift operator (1)
- Spatial isomorphism (1)
Articles 1 - 10 of 10
Full-Text Articles in Analysis
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Review: The Relationships Among Multiplicities Of A J-Self-Adjoint Differential Operator's Eigenvalue, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11
On The Norm Closure Problem For Complex Symmetric Operators, Stephan Ramon Garcia, Daniel E. Poore '11
Pomona Faculty Publications and Research
We prove that the set of all complex symmetric operators on a separable, infinite-dimensional Hilbert space is not norm closed.
Two Remarks About Nilpotent Operators Of Order Two, Stephan Ramon Garcia, Bob Lutz '13, D. Timotin
Two Remarks About Nilpotent Operators Of Order Two, Stephan Ramon Garcia, Bob Lutz '13, D. Timotin
Pomona Faculty Publications and Research
We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Unitary Equivalence To A Truncated Toeplitz Operator: Analytic Symbols, Stephan Ramon Garcia, Daniel E. Poore '11, William T. Ross
Pomona Faculty Publications and Research
Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators.
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.
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Spatial Isomorphisms Of Algebras Of Truncated Toeplitz Operators, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Pomona Faculty Publications and Research
We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Some New Classes Of Complex Symmetric Operators, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
We say that an operator $T \in B(H)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:H\to H$ 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 $(\dim \ker T, \dim \ker T^*)$.
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Complex Symmetric Partial Isometries, Stephan Ramon Garcia, Warren R. Wogen
Pomona Faculty Publications and Research
An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.
Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, And Similarity, Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Truncated Toeplitz Operators: Spatial Isomorphism, Unitary Equivalence, And Similarity, Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen
Pomona Faculty Publications and Research
A truncated Toeplitz operator A φ : K Θ → K Θ is the compression of a Toeplitz operator T φ : H 2 → H 2 to a model space K Θ ≔ H 2 ⊖ Θ H 2 . For Θ inner, let T Θ denote the set of all bounded truncated Toeplitz operators on K Θ . Our main result is a necessary and sufficient condition on inner functions Θ 1 and Θ 2 which guarantees that T Θ 1 and T Θ 2 are spatially isomorphic (i.e., U T Θ 1 = T Θ 2 U …
The Norm And Modulus Of A Foguel Operator, Stephan Ramon Garcia
The Norm And Modulus Of A Foguel Operator, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We develop a method for calculating the norm and the spectrum of the modulus of a Foguel operator. In many cases, the norm can be computed exactly. In others, sharp upper bounds are obtained. In particular, we observe several connections between Foguel operators and the Golden Ratio.