Abstract. We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to f , where f is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modi ed to produce inner functions.

The Toeplitz operator acting on the Bergman space A 2 (D), with symbol ϕ is given by Tϕf = P (ϕf ), where P is the projection from L 2 (D) onto the Bergman space. We present some history on the study of hyponormal Toeplitz operators acting on A 2 (D), as well as give results for when ϕ is a non-harmonic polynomial. We include a first investigation of Putnam's inequality for hyponormal operators with non-analytic symbols. Particular attention is given to unusual hyponormality behavior that arises due to the extension of the class of allowed symbols.where σ(T ) denotes the spectrum of T (cf. [2]).We study the hyponormality of certain operators acting on the Bergman spaceLet ϕ ∈ L ∞ (D). The Toeplitz operator T ϕ is given bywhere P is the orthogonal projection from L 2 (D) onto A 2 (D).2010 Mathematics Subject Classification. 47B35, 47B20.

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In [10], the authors have shown that Putnam's inequality for the norm of self-commutators can be improved by a factor of 1 2 for Toeplitz operators with analytic symbol ϕ acting on the Bergman space A 2 (Ω). This improved upper bound is sharp when ϕ(Ω) is a disk. In this paper we show that disks are the only domains for which the upper bound is attained.

Given a planar domain Ω, the Bergman analytic content measures the L 2 (Ω)-distance betweenz and the Bergman space A 2 (Ω). We compute the Bergman analytic content of simply-connected quadrature domains with quadrature formula supported at one point, and we also determine the function f ∈ A 2 (Ω) that best approximatesz. We show that, for simply-connected domains, the square of Bergman analytic content is equivalent to torsional rigidity from classical elasticity theory, while for multiply-connected domains these two domain constants are not equivalent in general.