Let $F\subset B(H)$ be a finite dimensional subvector space of the space of all bounded operators on a Hilbert space.

**Question:** Is there an upper bound for $$\{|tr(T)| \text{where} \quad T\in F\quad \text{is a trace class operator of unit norm}\}$$

An indirect but relevant motivation for this question is mentioned in the "Motivation" part of this post:

Irrational closed orbits of vector fields on $S^2$(Limit cycles and trace formula)

**Remark:** I have already learned from a specialist that the answer is "negative" if in this question we replace "trace class operators" with "Fredholm operators" and "trace" with "Fredholm index".