  We prove sharp upper bounds for sums of eigenvalues (and other spectral
functionals) of Laplace-like operators, including bi-Laplacian and fractional
Laplacian. We show that among linear images of a highly symmetric domain, our
spectral functionals are maximal on the original domain. We exploit the
symmetries of the domain, and the operator, avoiding necessity of finding good
test functions for variational problems. This is especially important for
fractional Laplacian, since exact solutions are not even known on intervals,
making it hard to find good test functions.
  To achieve our goals we generalize tight $p$-fusion frames, to extract the
best possible geometric results for domains with isometry groups admitting
tight $p$-frames. Any such group generates a tight $p$-fusion frame via
conjugation of arbitrary projection matrix. We show that generalized tight
$p$-frames can also be obtained by conjugation of arbitrary rectangular matrix,
with frame constant depending on the singular values of the matrix.
