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Register a new userRigidity, boundary interpolation and reproducing kernels
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We use reproducing kernel methods to study various rigidity problems. The methods and setting allow us to also consider the non-positive case.
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We study two geometric properties of reproducing kernels in model spaces $K_theta$where $theta$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that uniformly minimal non-Riesz$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $theta$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a zero localization property. In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.
For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H^2_beta$ for $1/2<beta<1$. Our main result is that, for a non-constant analytic function $varphi:DtoD$, the operator $C_{varphi }$ has dense range in $H_{beta }^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $varphi(D)$. It follows that if the range of $C_{varphi }$ is dense in $H_{beta }^{2}$, then $varphi $ is a weak-star generator of $H^{infty}$. Note that this conclusion is false for the classical Dirichlet space $mathfrak{D}$. We also characterize Fredholm composition operators on $H^{2}_{beta }$.
In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.
A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $kappa_1, ldots, kappa_N$, quaternions $p_1, ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u eq 1$ for $u = 1,ldots, N$, and quaternions $s_1, ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$lim_{substack{rrightarrow 1 rin(0,1)}} s(r p_u) = s_uquad {rm for} quad u=1,ldots, N,$$ and $$lim_{substack{rrightarrow 1 rin(0,1)}}frac{1-s(rp_u)overline{s_u}}{1-r}lekappa_u,quad {rm for} quad u=1,ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
In this paper we establish several invariant bounda
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