Spectra and resolvents

For a bounded operator A on a complex Hilbert space H, a complex number z belongs to the resolvent set when A−zI is bijective and its inverse is bounded on all of H. We use the convention R(z)=(A−zI)^{-1}; some texts reverse the sign. The spectrum is the complement of that set. For bounded A it is a nonempty compact subset of the complex plane. The bounded inverse theorem makes boundedness automatic once bijectivity between Banach spaces is established.

For a closed unbounded operator, the shifted map acts from its domain D(A) to H. A resolvent inverse must be defined on every vector of H, take values in D(A), and be bounded when regarded as a map H→H. The domain is part of the operator. Altering boundary conditions can change both the resolvent and spectrum even if the differential expression is unchanged. Later lessons will build that distinction rather than treating a derivative as a matrix with infinitely many rows.

Consider multiplication by x on L2(0,1). For z outside [0,1], division by x−z is bounded, so the shifted equation is uniquely solvable. At a real λ inside [0,1], an eigenfunction would have to vanish outside the singleton x=λ, hence represent zero in L2. Nevertheless functions concentrated near λ have unit norm while their shifted outputs become arbitrarily small. This rules out a bounded inverse and places every λ in [0,1] in the spectrum.

For this self-adjoint multiplication operator, those points are continuous spectrum: the shifted map is injective, its range is dense, and it is not onto. Density follows because the orthogonal complement of the range is the adjoint kernel, which is zero. Non-surjectivity can be checked with a right-hand side whose division by x−λ is not square integrable. This precise terminology describes a failure of stable inversion; it does not mean that an arbitrary operator's spectrum must be one continuous interval.

If |z| exceeds the norm of a bounded A, a Neumann series in A/z constructs the resolvent. This bounds the spectrum inside the norm disk. For a self-adjoint operator, the resolvent norm equals the reciprocal distance to the spectrum whenever z lies in the resolvent set. For a general nonnormal operator that exact equality fails: nearly dependent response directions can cause large amplification even when the spectral points are distant. This motivates pseudospectra in stability analysis.

A large resolvent does not by itself prove exponential growth of every time trajectory. It quantifies the sensitivity of a shifted solve; translating it into time-evolution statements needs additional assumptions and estimates. Similarly, numerical eigenvalues from a truncated matrix may miss resolvent amplification or approximate a continuous spectrum deceptively. Always state the operator, its domain, the chosen norm, and whether the claim concerns exact eigenvectors, approximate ones, or response to forcing.