A note on certain partial sum operators
Volume 73 / 2006
Banach Center Publications 73 (2006), 117-225
MSC: Primary 46L51; Secondary 42C10.
DOI: 10.4064/bc73-0-7
Abstract
We show that for the $t$-deformed semicircle measure, where $\frac{1}{2}< t\leq 1$, the expansions of $L_p$ functions with respect to the associated orthonormal polynomials converge in norm when $\frac{3}{2}< p< 3$ and do not converge when $1\leq p< \frac{3}{2}$ or $3< p$. From this we conclude that natural expansions in the non-commutative $L_p$ spaces of free group factors and of free commutation relations do not converge for $1\leq p< \frac{3}{2}$ or $3< p$.
