On the maximal projection constants

Barbara Lewandowska

Session: Optimal frames and codes (Dustin Mixon, Matthew Fickus)

Keywords: projection constansts, equiangular tight frames, Spherical (2, 2)-designs

Abstract:

Let $X$ be a Banach space over $\mathbb{K},$ where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}.$ Let $Y\subset X$ be a subspace.
By $\mathcal{P}(X,Y)$ denote the set of all linear and continuous projections from $X$ onto $Y$,
recalling that an operator $P \colon X \rightarrow Y$ is called a projection onto $Y$ if $P\mid_{Y}=Id_{Y}$
We define the relative projection constant of a subspace $Y$ of a space $X$ by
$$
\lambda(Y,X) :=\inf\lbrace\||P\||: P\in\mathcal{P}(X,Y)\rbrace.
$$
Now, we can define the absolute projection constant of $Y$ by
$$
\lambda(Y) :=\sup\lbrace\lambda(Y,X):Y\subset X\rbrace.
$$
The ultimate goal of researchers in this area is to determine the exact value of the maximal absolute projection constant, which is defined by
$$
\lambda_{\mathbb{K}}(m) :=\sup \lbrace\lambda(Y): \dim(Y)=m \rbrace.
$$
In 1960, B.Grünbaum conjectured that $\lambda_\mathbb{R}(2)=\frac{4}{3},$ and only in 2010, B. Chalmers and G. Lewicki proved it and that was the only known nontrivial case. Recently, we have provided exact values of $\lambda_{\mathbb{K}}(m)$ in cases where the maximal equiangular tight frame exists in $\mathbb{K}^m.$ There are numerous examples of complex maximal ETFs. In fact, it is conjectured that there is a complex maximal ETF in every dimension (Zaurner's conjecture).
Unlike in the complex case, real maximal ETFs seem to be rare objects. The only known cases are for $m$ equal to $2,$ $3,$ $7$ and $23.$ In other cases, the determination of the constant $\lambda_\mathbb{R}(m)$
seems to be difficult. Relying on the new construction of certain mutually unbiased equiangular tight frames,
we showed that $\lambda(5)\geq 5(11+6\sqrt{5})/59 \approx 2.06919.$
This value coincides with the numerical estimation of $\lambda(5)$ obtained by B. L. Chalmers,
thus reinforcing the belief that this is the exact value of $\lambda(5)$. We briefly discuss the above results in the talk and present conjectures based on some new concepts employing biangular tight frames and spherical (2,2)-designs.

Submission Number: 19