Zeros of hypergeometric functions and the norm of a composition operator. (English) Zbl 1111.47027

Hammond, Christopher

Comput. Methods Funct. Theory 6, No. 1, 37-50 (2006).

The authors consider the Hardy space \(H^2\) on the open unit disk \(\mathbb D\) in the complex plane, that is, the Hilbert space consisting of all analytic functions \(f(z) = \sum_{k=0}^\infty a_k z^k\) on \(\mathbb D\) such that \(\| f\| := (\sum_{k=0}^\infty | a_k| ^2)^{1/2} < \infty\). For any analytic map \(\varphi :\mathbb D\to\mathbb D\), the composition operator \(C_\varphi :H^2 \to H^2\) given by \(C_\varphi (f) = f \circ \varphi\) takes \(H^2\) boundedly into itself. It is well-known that
\[ {1 \over 1 - | \varphi (0)| ^2} \leq \| C_\varphi \| ^2 \leq {1 + | \varphi (0)| \over 1 - | \varphi (0)| }, \]
but the precise value of the norm \(\| C_\varphi \| \) is quite difficult to calculate. Nevertheless, some progress has been made in the case when \(\varphi\) is a linear fractional map. E. L. Basor and D. Q. Retsek [J. Math. Anal. Appl. 322, No. 2, 749–763 (2006; Zbl 1108.47024)] were able to demonstrate a connection between the norm of such an operator and the zeros of a particular hypergeometric series \(F(a,b,c;z) := \sum_{k=0}^\infty {(a)_k (b)_k \over (c)_k k!} z^k\), where \((d)_0 := 1\) and \((d)_k := d(d+1) \cdots (d+k-1)\) for \(k \geq 1\).
The authors pursue this line of inquiry further. They appeal to several results relating to hypergeometric series to deduce more information about the norm of a composition operator, in particular about the spectrum of \(C_\varphi^* C_\varphi\). For instance, the following theorem is proved: Let \(\alpha\) and \(\beta\) be real numbers with \(\delta = \alpha + \beta > 0\) and \(\beta - \alpha -1 > 0\), and consider the map \(\varphi (z) = {(\beta - 1)z + \alpha + 1 \over \alpha z + \beta}\). Then \(\sigma (C_\varphi^* C_\varphi ) = [0,\| C_\varphi \| ^2_e] \cup \{\lambda_k\}_{k=1}^m\), where \(\| C_\varphi \| _e\) is the essential norm of \(C_\varphi\), \(m = E(-\alpha + 1)\) (\(E(u) := 0\) if \(u \leq 0\); \(:= u - 1\) if \(u = 1,2,3,\dots\), and \(:=\) the smallest integer part of \(u\) if \(u > 0\) is not an integer) and \(\lambda_1, \dots ,\lambda_m\) are distinct eigenvalues greater that \(\| C_\varphi \| ^2_e\). Furthermore, for each \(\lambda_k\), the number \((\varphi '(1) \lambda_k)^{-1} = \| C_\varphi \| ^2_e \lambda_k^{-1}\) is a zero of \(F(\alpha , \beta , \delta ;z)\).
Furthermore, they use their knowledge of composition operators to establish the following result on the zeros of hypergeometric series: Let \(\alpha\) and \(\beta\) be complex numbers, with \(\delta = \overline{\alpha} + \beta > 0\), \(\beta - \alpha - 1 > 0\) and \(\alpha \not\in [0,+\infty )\). Then all of the zeros of \(F(\alpha , \beta , \delta ;z)\) within \(\mathbb D\) must lie on \((0,+\infty )\). Moreover, the smallest such zero \(x_0\) satisfies \[ {(\overline{\alpha} + \beta )(| \beta | - | \alpha + 1| ) \over (\beta - \alpha - 1)(| \beta | + | \alpha + 1| )} \leq x_0 \leq {(\overline{\alpha} + \beta )(| \beta | ^2 - | \alpha + 1| ^2) \over (\beta - \alpha - 1) | \beta | ^2}. \]

Reviewer: Luis Bernal Gonzales (Sevilla)

MSC:

47B33	Linear composition operators
33C05	Classical hypergeometric functions, \({}_2F_1\)
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D55	\(H^p\)-classes (MSC2000)
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

composition operator; operator norm; hypergeometric series

Citations:

Zbl 1108.47024

References:

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