  We introduce the new concepts of pseu\-do numerical range for operator
functions and families of sesquilinear forms as well as the pseu\-do block
numerical range for $n \times n$ operator matrix functions. While these notions
are new even in the bounded case, we cover operator polynomials with unbounded
coefficients, unbounded holomorphic form families of type (a) and associated
operator families of type (B). Our main results include spectral inclusion
properties of pseudo numerical ranges and pseudo block numerical ranges. For
diagonally dominant and off-diagonally dominant operator matrices they allow us
to prove spectral enclosures in terms of the pseudo numerical ranges of Schur
complements that no longer require dominance order $0$ and not even $<1$. As an
application, we establish a new type of spectral bounds for linearly damped
wave equations with possibly unbounded and/or singular damping.
