  We show that every graph continuous family of unbounded operators in a
Hilbert space becomes Riesz continuous after multiplication by an appropriate
family of unitary operators. This result provides a simple definition of the
index for graph continuous families of Fredholm operators and leads to two
corollaries for operators with compact resolvents: (1) the identity map between
the spaces of such operators equipped with the Riesz and the graph topology is
a homotopy equivalence; (2) every graph continuous family of such operators
acting between fibers of Hilbert bundles becomes Riesz continuous in
appropriate trivializations of the bundles.
  For self-adjoint operators, multiplication by unitary operators should be
replaced by conjugation. In general, a graph continuous family of self-adjoint
operators with compact resolvents cannot be made Riesz continuous by an
appropriate conjugation. We obtain a partial analogue of the trivialization
result above for self-adjoint operators and describe obstructions to existence
of such a trivialization arising in the general case. This part of results is
closely related to the recent work of N. Ivanov arXiv:2111.15081.
