# Trace class operators in the unit ball of a finite dimensional subvector space of $B(H)$

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".

If I understand the question correctly, the answer is yes, there is an upper bound. The set of trace class operators $$TC(H)$$ is itself a linear subspace of $$B(H)$$, so $$F' = F \cap TC(H)$$ is a finite dimensional linear subspace of $$TC(H)$$, and we are interested in $$\sup\{|{\rm tr}(T)|: T \in F', \|T\| = 1\}$$. Since $$F'$$ is finite dimensional, the restrictions of the trace norm and the operator norm to $$F'$$ are equivalent. This means that there is a constant $$C$$ such that $$|{\rm tr}(T)| \leq C\cdot \|T\|$$ for all $$T \in F'$$.