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

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 ${(\overline{\alpha} + \beta )(| \beta | - | \alpha + 1| ) \over (\beta - \alpha - 1)(| \beta | + | \alpha + 1| )} \leq x_0 \leq {(\overline{\alpha} + \beta )(| \beta | ^2 - | \alpha + 1| ^2) \over (\beta - \alpha - 1) | \beta | ^2}.$

### 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

Zbl 1108.47024
Full Text:

### References:

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