Finite matrices are controlled by eigenvalues and diagonalisation. Operator theory asks what survives when the vectors are functions or infinite sequences, the matrices become infinite-dimensional maps, and eigenvalues no longer describe the whole spectrum.
This guide develops the operator-theoretic core of R27.03. It begins with bounded operators on Banach and Hilbert spaces, defines the resolvent and spectrum, then separates point spectrum from the wider spectral set. It adds compact operators, adjoints, self-adjoint and normal structure, and closes with the domain discipline needed for unbounded operators.
Prerequisites: Normed and Banach Spaces and Hilbert Spaces and Orthogonal Projection.
Reading route: bounded operators → algebra and norm → invertibility → Neumann series → resolvent → spectrum → point spectrum → spectral radius → examples → adjoints → self-adjoint/unitary/normal operators → compact operators → compact spectral structure → spectral theorem → multiplication operators → unbounded operators and domains → practice and solutions.
1. A bounded operator is an infinite-dimensional linear transformation with analytic control
For normed spaces X,Y, a linear map T:X→Y is bounded if
||Tx||≤C||x||
for some C and every x.
The smallest such C is the operator norm ||T||. Boundedness is equivalent to continuity for linear maps.
2. B(X) is a Banach algebra when X is Banach
Let B(X)=B(X,X). Operators can be added, scaled and composed.
The operator norm satisfies
||ST||≤||S||||T||.
If X is Banach, then B(X) is complete in the operator norm. Composition therefore lives inside a complete normed algebra.
3. Invertibility must include a bounded inverse
For a bounded operator T:X→X on a Banach space, the analytically useful notion is bounded invertibility: T is bijective and T^{-1} is bounded.
By the bounded inverse theorem, a bounded linear bijection between Banach spaces automatically has bounded inverse.
4. The Neumann series gives a perturbative inverse
If ||A||<1 in a Banach algebra, then
(I−A)^{-1}=Σ_{n=0}^∞ A^n.
The series converges in operator norm because Σ||A||^n converges.
This proves that invertibility is stable under sufficiently small operator-norm perturbations.
5. The resolvent asks where T−λI is invertible
For T∈B(X) over C, the resolvent set is
ρ(T)={λ∈C:T−λI is boundedly invertible}.
The resolvent operator is
R(λ,T)=(T−λI)^{-1}
for λ∈ρ(T).
6. The spectrum is the failure set of bounded invertibility
The spectrum is
σ(T)=C\ρ(T).
In finite dimensions, spectrum equals the eigenvalue set. In infinite dimensions, spectrum can contain values that are not eigenvalues.
This is one of the decisive breaks from matrix intuition.
7. Eigenvalues form the point spectrum
λ is an eigenvalue if some nonzero x satisfies
Tx=λx.
Equivalently T−λI is not injective.
The set of eigenvalues is called the point spectrum σ_p(T), and σ_p(T)⊂σ(T).
8. Spectrum can occur without eigenvectors
On L²([0,1]), let M_x be multiplication by x:
(M_xf)(t)=t f(t).
Its spectrum is [0,1]. Yet for any fixed λ∈[0,1], the equation (t−λ)f(t)=0 forces f to be supported on one point, hence f=0 as an L² element.
So M_x has spectrum [0,1] but no eigenvalues. Infinite-dimensional spectrum contains more than point spectrum.
9. The spectrum of a bounded complex operator is nonempty and compact
For T∈B(X) on a nonzero complex Banach space, σ(T) is a nonempty compact subset of C.
Moreover
σ(T)⊂{λ:|λ|≤||T||}.
For |λ|>||T||, factor T−λI=−λ(I−T/λ) and invert using the Neumann series.
10. The spectral radius measures the outer spectral scale
Define
r(T)=sup{|λ|:λ∈σ(T)}.
The spectral-radius formula states
r(T)=lim_{n→∞}||T^n||^{1/n}=inf_n||T^n||^{1/n}.
Always r(T)≤||T||, sometimes strictly.
11. Nilpotent operators show norm and spectrum are different quantities
If T^m=0 for some m, then σ(T)={0}, so r(T)=0.
Yet ||T|| can be positive and large.
Spectral information describes long-term algebraic growth differently from one-step operator norm.
12. Resolvent identities connect different spectral parameters
If λ,μ∈ρ(T), then
R(λ,T)−R(μ,T)=(λ−μ)R(λ,T)R(μ,T)
up to the sign convention determined by whether one uses (T−λI)^{-1} or (λI−T)^{-1}.
The resolvent is analytic on ρ(T), turning operator inversion into a complex-analytic object.
13. On Hilbert space every bounded operator has an adjoint
For T∈B(H), Riesz representation gives a unique T*∈B(H) satisfying
⟨Tx,y⟩=⟨x,T*y⟩.
The adjoint obeys (ST)*=T*S*, (T*)*=T and ||T*||=||T||.
14. Self-adjoint operators generalise real symmetric matrices
T is self-adjoint when T=T*.
For bounded self-adjoint T:
- σ(T)⊂R;
- ||T||=sup_{||x||=1}|⟨Tx,x⟩|;
- spectral decomposition behaves like infinite-dimensional diagonalisation.
Self-adjointness is stronger than merely having real eigenvalues.
15. Unitary operators generalise orthogonal rotations
U is unitary if
U*U=UU*=I.
Then ||Ux||=||x|| and σ(U) lies on the unit circle.
Unitary maps preserve the full Hilbert-space geometry.
16. Normal operators are the natural spectral-theorem class
T is normal if
T*T=TT*.
Self-adjoint and unitary operators are normal.
For bounded normal operators, the spectral theorem gives a projection-valued representation rather than necessarily a countable eigenvector basis.
17. Compact operators turn bounded sets into relatively compact sets
A bounded operator K:X→Y is compact if it maps the unit ball, equivalently every bounded sequence, into a set whose closure is compact, equivalently produces convergent subsequences from bounded sequences after applying K.
Finite-rank operators are compact. Norm limits of compact operators are compact.
18. Integral operators often provide compact examples
On suitable function spaces, an operator
(Kf)(x)=∫ k(x,y)f(y)dy
with a sufficiently regular square-integrable kernel can be compact.
Compactness reflects smoothing or compression: bounded input families are sent into sets with stronger sequential control.
19. Compact-operator spectra resemble matrix spectra away from zero
For a compact operator K on an infinite-dimensional Banach space, every nonzero spectral value is an eigenvalue.
Nonzero eigenvalues have finite algebraic/geometric multiplicity in appropriate formulations, and any infinite sequence of distinct nonzero eigenvalues can accumulate only at 0.
Zero is the special infinite-dimensional spectral point.
20. Compact self-adjoint operators recover an orthonormal eigenbasis
If K is compact and self-adjoint on a Hilbert space, H decomposes into the orthogonal sum of Ker(K) and eigenspaces for real nonzero eigenvalues.
Under separability, one obtains an orthonormal eigenbasis for the closure of the range together with a basis of the kernel.
This is the closest infinite-dimensional analogue of diagonalising a real symmetric matrix.
21. The bounded normal spectral theorem uses projection-valued measures
For bounded normal T on a complex Hilbert space, there exists a projection-valued measure E on σ(T) such that
T=∫_{σ(T)} λ dE(λ).
More generally, bounded Borel functions φ define operators φ(T)=∫φ(λ)dE(λ).
This functional calculus replaces finite diagonal matrices by continuous spectral data when eigenvectors alone are insufficient.
22. Multiplication operators model continuous spectrum transparently
On L², let (M_mf)(x)=m(x)f(x) for bounded measurable m.
Then
||M_m||=||m||∞
and the spectrum is the essential range of m.
The spectral theorem says every bounded normal operator is, in an appropriate representation, multiplication-like.
23. The unilateral shift shows spectrum need not equal eigenvalues
On ℓ²(N), define S(x₁,x₂,…)= (0,x₁,x₂,…).
S is an isometry but not onto. It has no eigenvalues, yet its spectrum is the closed unit disk.
Its adjoint S* has many eigenvalues inside the unit disk.
This example decisively separates spectrum, point spectrum and surjectivity.
24. Spectral projections separate parts of the spectrum
For normal operators, the spectral measure E assigns orthogonal projections to measurable spectral sets.
If Δ₁ and Δ₂ are disjoint, their spectral projections have orthogonal ranges.
This lets one decompose H according to spectral regions rather than only individual eigenvalues.
25. Functional calculus turns scalar functions into operator functions
If T is normal, one can define φ(T) for broad classes of φ on σ(T).
Examples include powers, exponentials and indicator functions of spectral sets.
This is the rigorous basis for expressions such as e^{itT} in evolution equations when the operator hypotheses are appropriate.
26. Unbounded operators require an explicit domain
Differentiation on L² is not a bounded operator defined on all of L².
An unbounded operator is therefore written
T:D(T)⊂H→H
with a specified domain D(T).
Changing the domain can change closedness, adjoints, self-adjointness and spectrum even when the formal differential expression is unchanged.
27. Symmetric is not the same as self-adjoint for unbounded operators
An operator T is symmetric when
⟨Tx,y⟩=⟨x,Ty⟩
for x,y in D(T).
Self-adjointness additionally requires T=T* with equality of domains D(T)=D(T*).
This domain equality is essential in differential operators and quantum mechanics.
28. Closed operators preserve limits of graph pairs
T:D(T)→Y is closed if x_n→x and Tx_n→y imply x∈D(T) and Tx=y.
Unbounded self-adjoint operators are closed. Densely defined closable operators can be replaced by their graph closure.
Closedness is the correct stability condition for many unbounded operators.
29. Common theorem-boundary errors
- In infinite dimensions, spectrum need not equal eigenvalues.
- Algebraic invertibility is not enough unless bounded inverse control is established.
- The spectrum of a bounded complex operator is nonempty; this statement uses complex Banach-space structure.
- Compact does not mean finite rank.
- Compact-operator matrix-like spectral structure applies away from zero.
- Self-adjoint and symmetric are not interchangeable for unbounded operators.
- An unbounded operator is not specified until its domain is specified.
- Normal spectral decomposition may require projection-valued continuous spectral data rather than an eigenbasis.
30. A dependable spectral workflow
State the space, scalar field and operator domain first. Determine whether the operator is bounded, compact, self-adjoint, unitary or normal before selecting spectral tools.
To test λ, inspect T−λI for injectivity, dense/surjective range and bounded inverse. Do not search only for eigenvectors.
For compact operators, use eigenvalue structure away from zero. For normal operators, use spectral measures and functional calculus. For differential operators, keep the domain visible at every step.
31. Independent practice: twenty questions
- Define a bounded linear operator.
- State the operator-norm submultiplicative inequality.
- When does the Neumann series invert I−A?
- Define the resolvent set.
- Define the spectrum.
- Define the point spectrum.
- Why can spectrum contain non-eigenvalues?
- Where does σ(T) lie relative to ||T||?
- Define the spectral radius.
- State the spectral-radius formula.
- Define the adjoint T*.
- Define self-adjoint, unitary and normal.
- Where does the spectrum of a bounded self-adjoint operator lie?
- Where does the spectrum of a unitary operator lie?
- Define a compact operator.
- What is special about nonzero spectral values of a compact operator?
- State the compact self-adjoint spectral-theorem idea.
- What is the spectrum of multiplication by m on L²?
- Why must an unbounded operator include a domain?
- Why is symmetric weaker than self-adjoint for unbounded operators?
32. Worked solutions and checks
1. Linear T with ||Tx||≤C||x|| for all x.
2. ||ST||≤||S||||T||.
3. When ||A||<1, with inverse ΣA^n.
4. ρ(T) is the set of λ for which T−λI is boundedly invertible.
5. σ(T)=C\ρ(T).
6. Eigenvalues: λ such that T−λI is not injective.
7. Failure of surjectivity or bounded inverse can occur even when no eigenvector exists.
8. σ(T) lies in the closed disk |λ|≤||T||.
9. r(T)=sup{|λ|:λ∈σ(T)}.
10. r(T)=lim||T^n||^{1/n}.
11. ⟨Tx,y⟩=⟨x,T*y⟩.
12. Self-adjoint: T=T*. Unitary: T*T=TT*=I. Normal: T*T=TT*.
13. On the real axis.
14. On the unit circle.
15. A bounded operator mapping bounded sequences to sequences with convergent image subsequences.
16. Every nonzero spectral value is an eigenvalue; distinct nonzero eigenvalues can accumulate only at 0.
17. A compact self-adjoint operator decomposes into orthogonal eigenspaces for real eigenvalues, with only 0 as possible accumulation point.
18. The essential range of m.
19. The formal rule may not act on every vector, and domain choice affects adjoints, closedness and spectrum.
20. Self-adjointness requires equality with the adjoint including equality of domains, not just equality of inner products on D(T).
33. Where this guide hands off
R27.04 turns Hilbert and operator structure into harmonic decomposition. Fourier series become orthonormal expansions, convolution becomes multiplication in frequency, translation becomes phase, and Parseval turns signal energy into coefficient energy.
Sources and further study
See MIT OpenCourseWare 18.102 Introduction to Functional Analysis for operators, spectrum, compactness and Hilbert-space spectral theory.