  Pseudo-Hermitian operators generalize the concept of Hermiticity. This class
of operators includes the quasi-Hermitian operators, which reformulate quantum
theory while retaining real-valued measurement outcomes and unitary time
evolution. This thesis is devoted to the study of locality in quasi-Hermitian
theory, the symmetries and conserved quantities associated with non-Hermitian
operators, and the perturbative features of pseudo-Hermitian matrices. In
addition to the presented original research, scholars will appreciate the
lengthy introduction to non-Hermitian physics.
  Local quasi-Hermitian observable algebras are examined. Expectation values of
local quasi-Hermitian observables equal expectation values of local Hermitian
observables. Thus, quasi-Hermitian theories do not increase the values of
nonlocal games set by Hermitian theories. Furthermore, Bell's inequality
violations in quasi-Hermitian theories never exceed the Tsirelson bound of
Hermitian quantum theory.
  Exceptional points, which are branch points in the spectrum, are a
perturbative feature unique to non-Hermitian operators. Cusp singularities of
algebraic curves are related to higher-order exceptional points.
  To exemplify novelties of non-Hermiticity, one-dimensional lattice models
with a pair of non-Hermitian defect potentials with balanced loss and gain,
$\Delta \pm i \gamma$, are explored. When the defects are nearest neighbour,
the entire spectrum becomes complex when $\gamma$ is tuned past a second-order
exceptional point. When the defects are at the edges of the chain and the
hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the
$\mathcal{PT}$-phase transition is dictated by the topological phase.
  Chiral symmetry and representation theory are used to derive large classes of
pseudo-Hermitian operators with closed-form intertwining operators.
