On the Operators with Numerical Range in an Ellipse
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911909619433472 |
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| author | Agler, Jim Lykova, Zinaida A. Young, N. J. |
| author_facet | Agler, Jim Lykova, Zinaida A. Young, N. J. |
| contents | We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_δ\subseteq \mathbb{C}$ given by \[ K_δ{\stackrel{\rm def}{=}}
\left\{x+iy: \frac{x^2}{(1+δ)^2} + \frac{y^2}{(1-δ)^2} \leq 1\right\}, \] where $0 < δ< 1$. Here $\mathcal{B}(\mathcal{H})$ denotes the collection of bounded linear operators on a Hilbert space $\mathcal{H}$. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem.
We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators $T$ that satisfy appropriate dilation properties. This generalization yields a characterization of the operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in $K_δ$ in terms of certain structured contractions that act on $\mathcal{H} \oplus \mathcal{H}$.
As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those $T$ such that $W(T)\subseteq K_δ$. We prove that, if $T$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, then $W(T)\subseteq K_δ$ if and only if there exist a pair of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is self-adjoint and \[ T=2\sqrtδA + (1-δ)\sqrt{1+A}\ B\sqrt{1-A}. \] We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal $H^\infty$-extension problem for analytic functions defined on a slice of the symmetrized bidisc. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_00680 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Operators with Numerical Range in an Ellipse Agler, Jim Lykova, Zinaida A. Young, N. J. Functional Analysis 47A12, 15A60, 47B99 We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_δ\subseteq \mathbb{C}$ given by \[ K_δ{\stackrel{\rm def}{=}} \left\{x+iy: \frac{x^2}{(1+δ)^2} + \frac{y^2}{(1-δ)^2} \leq 1\right\}, \] where $0 < δ< 1$. Here $\mathcal{B}(\mathcal{H})$ denotes the collection of bounded linear operators on a Hilbert space $\mathcal{H}$. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators $T$ that satisfy appropriate dilation properties. This generalization yields a characterization of the operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in $K_δ$ in terms of certain structured contractions that act on $\mathcal{H} \oplus \mathcal{H}$. As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those $T$ such that $W(T)\subseteq K_δ$. We prove that, if $T$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, then $W(T)\subseteq K_δ$ if and only if there exist a pair of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is self-adjoint and \[ T=2\sqrtδA + (1-δ)\sqrt{1+A}\ B\sqrt{1-A}. \] We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal $H^\infty$-extension problem for analytic functions defined on a slice of the symmetrized bidisc. |
| title | On the Operators with Numerical Range in an Ellipse |
| topic | Functional Analysis 47A12, 15A60, 47B99 |
| url | https://arxiv.org/abs/2311.00680 |