I am working with certain submanifolds of symmetric spaces and, using a construction in Terng-Thorbergson, we ended up in the following Hilbert space problem:
Let $H$ be a (real) Hilbert Space and $T_t:H \rightarrow H$ be a smooth $1$-parameter family of symmetric operators. Suppose that there is a finite-dimensional subespace $V$ of $H$ such that the spectrum of both $T_t$ and $T_t|_{V^\perp}$ does not depend on $t$. Is it true that the trace of $T_t|_V$ also does not depend on $t$?