On uniformly bounded orthonormal Sidon systems

In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any uniformly bounded $\psi_2 (C)$-orthonormal system ($\psi_2 (C)$ is a variant of subGaussian)is 2-fold tensor Sidon. This sharpens their result that it is 5-fold tensor Sidon. The proof is somewhat reminiscent of the author's original one for (Abelian) group characters, based on ideas due to Drury and Rider. However, we use Talagrand's majorizing measure theorem in place of Fernique's metric entropy lower bound. We also show that a uniformly bounded orthonormal system is randomly Sidon iff it is 4-fold tensor Sidon, or equivalently $k$-fold tensor Sidon for some (or all) $k\ge 4$. Various generalizations are presented, including the case of random matrices, for systems analogous to the Peter-Weyl decomposition for compact non-Abelian groups. In the latter setting we also include a new proof of Rider's unpublished result that randomly Sidon sets are Sidon, which implies that the union of two Sidon sets is Sidon.Comment: v3: randomly Sidon implies four-fold tensor Sidon. v6: preceding is extended to matrix valued case, also an illustrative-hopefully illuminating-example is presented. Terminolgy is improve

Multipliers of the Hardy space H^1 and power bounded operators

We study the space of functions \phi\colon \NN\to \CC such that there is a Hilbert space $H$, a power bounded operator $T$ in $B(H)$ and vectors $\xi,\eta$ in $H$ such that $$\phi(n) = .$$ This implies that the matrix $(\phi(i+j))_{i,j\ge 0}$ is a Schur multiplier of $B(\ell_2)$ or equivalently is in the space (\ell_1 \buildrel {\vee}\over {\otimes} \ell_1)^*. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of $H^1$ which we call ``shift-bounded''. We show that there is a $\phi$ which is a ``completely bounded'' multiplier of $H^1$, or equivalently for which $(\phi(i+j))_{i,j\ge 0}$ is a bounded Schur multiplier of $B(\ell_2)$, but which is not ``shift-bounded'' on $H^1$. We also give a characterization of ``completely shift-bounded'' multipliers on $H^1$.Comment: Submitted to Colloquium Mat

Quantum expanders and growth of group representations

Let $\pi$ be a finite dimensional unitary representation of a group $G$ with a generating symmetric $n$-element set $S\subset G$. Fix \vp>0. Assume that the spectrum of $|S|^{-1}\sum_{s\in S} \pi(s) \otimes \overline{\pi(s)}$ is included in [-1, 1-\vp] (so there is a spectral gap \ge \vp). Let $r'_N(\pi)$ be the number of distinct irreducible representations of dimension $\le N$ that appear in $\pi$. Then let R_{n,\vp}'(N)=\sup r'_N(\pi) where the supremum runs over all $\pi$ with {n,\vp} fixed. We prove that there are positive constants \delta_\vp and c_\vp such that, for all sufficiently large integer $n$ (i.e. $n\ge n_0$ with $n_0$ depending on \vp) and for all $N\ge 1$, we have \exp{\delta_\vp nN^2} \le R'_{n,\vp}(N)\le \exp{c_\vp nN^2}. The same bounds hold if, in $r'_N(\pi)$, we count only the number of distinct irreducible representations of dimension exactly $= N$.Comment: Main addition: A remark due to Martin Kassabov showing that the numbers R(N) grow faster than polynomial. v3: Minor clarification