Similarity of operators and geometry of eigenvector bundles.

*(English)*Zbl 1190.47002From the authors’ introduction: “The main objects of this paper are operators with complete analytic families of eigenvectors, the backward shift being one of the simplest examples of such operators. Characterization of such operators up to unitary equivalence was completely done by M. J. Cowen and R. G. Douglas in [Acta Math. 141, 187–261 (1978; Zbl 0427.47016)].”

Let \(H\) be a Hilbert space and let \(T\in \mathcal B(H)\) be a contraction. Assume further that \(\dim\text{Ker}\,(T-\lambda I)=n<\infty\) for all \(\lambda\) in the unit disk \(\mathbb D,\) that \(\text{span}\,\cup_{\lambda \in \mathbb D}\text{Ker}\,(T-\lambda I)=H\) and that \(\text{Ker}\,(T-\lambda I)\) depends analytically on \(\lambda\).

On the vector Hardy space \(H_n^2\), the backward shift \(S^*_nB\) is defined in a similar way as is defined on \(H^2\); namely, \(S^*_nf(z)=\frac{f(z)-f(0)}{z}\).

For a contraction \(T\) as above, the authors prove that four statements are equivalent. The precise form of (2), (3) and (4) is very technical, but roughly, one finds:

(1) \(T\) is similar to \(S^*_n\).

(2) The eigenvector bundles of \(T\) and \(S_n^*\) are “uniformly equivalent”.

(3) The Green potential of an expression containing the mean curvature of the eigenvector bundle of \(T\) is uniformly bounded inside the disk \(\mathbb D\).

(3) A measure related to the mean curvature of the eigenvector bundle of \(T\) is a Carleson measure.

The equivalence of (1) and (2) was obtained by M. Uchiyama [Trans. Am. Math. Soc. 319, No. 1, 405–415 (1990; Zbl 0733.47015)] who also considered quasisimilarity, while B. Sz.-Nagy and C. Foiaş had characterized operators similar to isometries in terms of their characteristic functions in [Acta Sci. Math. 26, 79–91 (1965; Zbl 0138.38905)].

Let \(H\) be a Hilbert space and let \(T\in \mathcal B(H)\) be a contraction. Assume further that \(\dim\text{Ker}\,(T-\lambda I)=n<\infty\) for all \(\lambda\) in the unit disk \(\mathbb D,\) that \(\text{span}\,\cup_{\lambda \in \mathbb D}\text{Ker}\,(T-\lambda I)=H\) and that \(\text{Ker}\,(T-\lambda I)\) depends analytically on \(\lambda\).

On the vector Hardy space \(H_n^2\), the backward shift \(S^*_nB\) is defined in a similar way as is defined on \(H^2\); namely, \(S^*_nf(z)=\frac{f(z)-f(0)}{z}\).

For a contraction \(T\) as above, the authors prove that four statements are equivalent. The precise form of (2), (3) and (4) is very technical, but roughly, one finds:

(1) \(T\) is similar to \(S^*_n\).

(2) The eigenvector bundles of \(T\) and \(S_n^*\) are “uniformly equivalent”.

(3) The Green potential of an expression containing the mean curvature of the eigenvector bundle of \(T\) is uniformly bounded inside the disk \(\mathbb D\).

(3) A measure related to the mean curvature of the eigenvector bundle of \(T\) is a Carleson measure.

The equivalence of (1) and (2) was obtained by M. Uchiyama [Trans. Am. Math. Soc. 319, No. 1, 405–415 (1990; Zbl 0733.47015)] who also considered quasisimilarity, while B. Sz.-Nagy and C. Foiaş had characterized operators similar to isometries in terms of their characteristic functions in [Acta Sci. Math. 26, 79–91 (1965; Zbl 0138.38905)].

Reviewer: Héctor N. Salas (Mayagüez)

##### MSC:

47A10 | Spectrum, resolvent |

47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |

30H10 | Hardy spaces |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

##### Keywords:

similarity; curvature of the eigenvector bundle; backward shift; Carleson measure; reproducing kernel
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\textit{H.-K. Kwon} and \textit{S. Treil}, Publ. Mat., Barc. 53, No. 2, 417--438 (2009; Zbl 1190.47002)

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