The failure of rational dilation on a triply connected domain

Rational dilation for operators associated with spectral interpolation and distinguished varieties

The main aims of this article are to characterize a class of operators associated with the symmetrized polydisc that admit rational dilations on the minimal space and to show an interplay between rational dilation and distinguished varieties in the symmetrized polydisc.

Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains

We show that if R is a compact domain in the complex plane with two or more holes and an anticonformal involution onto itself (or equivalently a hyperelliptic Schottky double), then there is an operator T which has R as a spectral set, but does not dilate to a normal operator with spectrum on the boundary of R.

The failure of rational dilation on the tetrablock

We show by a counter example the failure of rational dilation on the tetrablock, a polynomially convex and non-convex domain in $\mathbb C^3$, defined as $$ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever } |z|\leq 1, |w|\leq 1 \}. $$ A commuting triple of operators $(T_1,T_2,T_3)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set, is called an $\mathbb E$-contraction. For an $\mathbb E$-contraction $(T_1,T_2,T_3)$, the two operator equations $$ T_1-T_2^*T_3=D_{T_3}X_1D_{T_3} \textup{ and } T_2-T_1^*T_3= D_{T_3}X_2D_{T_3}, \quad …

Rational dilation problems associated with constrained algebras

It is shown that rational dilation fails on broad collection of distinguished varieties associated to constrained subalgebras of the disk algebra of the form C + B A(D), where B is a finite Blaschke product with two or more zeros. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler-Pick interpolation theorem is given and it is proved that there exist Kaijser-Varopoulos style examples of non-contractive unital representations where the generators are contractions.

Dilation theory in finite dimensions and matrix convexity

We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This result can be seen as a general principle to deduce finite-dimensional dilation theorems from their classical infinite-dimensional counterparts. In addition to providing unified proofs of known finite-dimensional dilation theorems, we establish finite-dimensional versions of Agler's theorem on rational dilation on an annulus, of Berger's dilation theorem for operators of numerical radius at most $1$, and of the Putinar-Sandberg numerical range dila…

L2(R)的有理正交Littlewood-Paley小波基

Rational Orthonormal Littlewood-Paley Wavelet Basis of L2(R)

本文证明Auscher提出的非整数有理伸缩因子Littlewood-Paley小波非正交基，进而提出对所有有理数伸缩因子成立的新正交基，并通过带通信号采样定理完成证明。