Singular values and the spectrum of an operator Are the singular values of a bounded linear operator on an Hilbert space in the spectrum of a that operator in general? If not, is there a subclass of operators for which this hold?

Singular values are the same for an operator $A$ and any unimodular multiple $\alpha A$, with $\alpha\in\mathbb C$. On the other hand, multiplication by $\alpha$ rotates the spectrum of $A$. This leads to an exaemple, given by PhoemueX, of $x\mapsto -x$, with singular value $1$ and spectrum $\\{-1\\}$.

> is there a subclass of operators for which this hold?

For positive self-adjoint operators the singular values are in the spectrum, since they are the eigenvalues of $\sqrt{A^*A}=A$.