  Aim of this paper is trying to show the possible significance, and
usefulness, of various non-selfadjoint operators for suitable Observables in
non-relativistic and relativistic quantum mechanics, and in quantum
electrodynamics: More specifically, this work starts dealing with: (i) the
hermitian (but not selfadjoint) Time operator in non-relativistic quantum
mechanics and in quantum electrodynamics; with (ii) idem, with the introduction
of Time and Space operators; and with (iii) the problem of the four-position
and four-momentum operators, each one with its hermitian and anti-hermitian
parts, for relativistic spin-zero particles. Afterwards, other physical
applications of non-selfadjoint (and even non-hermitian) operators are briefly
discussed. We mention how non-hermitian operators can indeed be used in physics
[as it was done, elsewhere, for describing Unstable States]; and some
considerations are added on the cases of the nuclear optical potential, of
quantum dissipation, and in particular of an approach to the measurement
problem in QM in terms of a "chronon". [This chapter is largely based on work
developed, along the years, in collaboration with V.S.Olkhovsky, and, in
smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk]. PACS
numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 03.65.Yz; 11.10.St;
11.10.-z; 11.90.+t; 02.00.00; 03.00.00; 24.10.Ht; 03.65.Yz; 21.60.-u; 11.10.Ef;
03.65.Fd; 02.40.Dr; 98.80.Jk. Keywords: time operator, space-time operator,
non-selfadjoint operators, non-hermitian operators, bilinear operators, time
operator for discrete energy spectra, time-energy uncertainty relations,
Klein-Gordon equation, chronon, quantum dissipation, decoherence, nuclear
optical model, cosmology, projective relativity.
