Unbounded Operators 8 | Adjoint Operators

TL;DR

For bounded operators on Hilbert spaces, the adjoint T* is characterized by ⟨y, Tx⟩ = ⟨T*y, x⟩ for all x and y.

Briefing Cornell Notes

Briefing

Adjoint operators for unbounded operators are defined by shifting the “push to the other side” idea from bounded operators, but the price is new domain conditions. In Hilbert spaces, the adjoint of an unbounded operator is built so that inner products match after moving the operator from one slot to the other; in Banach spaces, the same principle is implemented using continuous linear functionals on dual spaces. The key takeaway is that these adjoints exist only under a density assumption, and when they do exist they come with a maximal domain—so there’s a unique “largest” adjoint consistent with the defining identity.

For bounded operators on Hilbert spaces, the adjoint T* is characterized by the identity ⟨y, Tx⟩ = ⟨T*y, x⟩ for all x and y. This definition relies on inner products and therefore doesn’t directly transfer to Banach spaces, where there is no inner product structure to “move across.” Instead, Banach-space adjoints use the dual pairing: for a bounded operator T: X → Y, the adjoint T′ maps Y′ → X′ and is defined by requiring that for every y′ ∈ Y′ and x ∈ X, (T′y′)(x) = y′(Tx). This turns the operator into a transformation between spaces of continuous linear functionals, and in Banach spaces the dual-space framework is what replaces inner products.

Extending to unbounded operators introduces domains as part of the definition. In the Banach-space setting, start with a linear operator T with domain D(T) ⊂ X, and assume T is densely defined—meaning D(T) is dense in X. Under this assumption, one can define an adjoint operator T′ with a domain contained in Y′, chosen so that the defining relation holds: for y′ in the adjoint’s domain and x in D(T), (T′y′)(x) = y′(Tx). The density of D(T) is crucial for two reasons: it ensures the adjoint is well defined, and it supports uniqueness. If two candidate functionals on X′ agree on the dense set D(T), continuity forces them to agree everywhere, so the adjoint’s action cannot depend on arbitrary choices. The result is a uniquely determined adjoint with a maximal domain—there’s no larger domain on which the same identity can consistently hold.

In Hilbert spaces, the unbounded adjoint T* is defined analogously but uses vectors instead of functionals. For a densely defined operator T: D(T) ⊂ X → Y, the adjoint T* maps from a subset of Y into X and is characterized by the requirement that ⟨y, Tx⟩ = ⟨T*y, x⟩ for all x ∈ D(T). As in the Banach case, dense definition is the condition that makes the adjoint exist and be well defined, and the adjoint comes with a largest possible domain consistent with the identity.

A final caution matters: even when the adjoint exists, its domain might be too small to be useful. In the worst case it could collapse to just the zero vector, meaning T* would carry little information about T. Determining when the adjoint’s domain is sufficiently large is flagged as the next step for the series, underscoring that existence alone isn’t the whole story.

Cornell Notes

Unbounded adjoints are built by enforcing the same “move the operator across” identity used for bounded operators, but domains must be included. In Banach spaces, a densely defined operator T: D(T) ⊂ X → Y has a Banach-space adjoint T′ mapping into X′, defined so that (T′y′)(x) = y′(Tx) for y′ in the adjoint’s domain and x ∈ D(T). Dense definition ensures the adjoint is well defined and unique, because agreement on a dense subset forces equality everywhere by continuity. In Hilbert spaces, the adjoint T* is defined similarly using inner products: ⟨y, Tx⟩ = ⟨T*y, x⟩ for x ∈ D(T). Even then, the adjoint’s domain may be too small, so later discussion focuses on when it becomes informative.

Why does the adjoint definition for Banach spaces rely on dual spaces rather than inner products?

Banach spaces lack an inner product, so there’s no direct way to express “⟨y, Tx⟩ = ⟨T*y, x⟩.” Instead, the adjoint is defined through continuous linear functionals. For bounded T: X → Y, the Banach-space adjoint T′ maps Y′ → X′ and satisfies (T′y′)(x) = y′(Tx) for every y′ ∈ Y′ and x ∈ X. This uses the dual pairing to replicate the bounded Hilbert-space identity in a setting where inner products aren’t available.

What role does dense definition of an unbounded operator play in making the adjoint well defined?

For an unbounded operator T with domain D(T) ⊂ X, dense definition means D(T) is dense in X. The adjoint identity is required only for x ∈ D(T), not for all x ∈ X. Dense definition ensures that if two candidate functionals on X′ agree on D(T), continuity forces them to agree on all of X. That prevents ambiguity and guarantees the adjoint is well defined.

How does uniqueness of the unbounded Banach-space adjoint follow from density?

Suppose two functionals X1′ and X2′ both satisfy the adjoint identity and therefore agree on all x ∈ D(T). Then (X1′ − X2′)(x) = 0 on the dense set D(T). Because the difference is a continuous linear functional, vanishing on a dense subset implies it vanishes everywhere. Hence X1′ = X2′, giving uniqueness. The adjoint’s domain is therefore maximal: there’s no larger domain where the identity can hold without contradiction.

What is the Hilbert-space adjoint identity for unbounded operators?

For a densely defined unbounded operator T: D(T) ⊂ X → Y, the adjoint T* is defined on those y ∈ Y for which there exists an element T*y ∈ X satisfying ⟨y, Tx⟩ = ⟨T*y, x⟩ for every x ∈ D(T). This mirrors the bounded Hilbert-space characterization but restricts the equality to the operator’s domain D(T).

Why might the adjoint’s domain be too small to be useful even when it exists?

The adjoint is defined only for those y (or y′) that make the defining identity work with a corresponding element in the target space. In principle, the set of such y can be minimal. The transcript flags the possibility that the adjoint’s domain could be as small as just the zero vector, which would make T* carry little or no information about T. Later discussion is promised on conditions that ensure a sufficiently large adjoint domain.

Review Questions

In Banach spaces, what exact equation defines T′, and on which sets of x and y′ must it hold?
How does dense definition of D(T) in X guarantee uniqueness of the unbounded adjoint?
In Hilbert spaces, what does it mean for a vector y to belong to the domain of T*?

Key Points

1
For bounded operators on Hilbert spaces, the adjoint T* is characterized by ⟨y, Tx⟩ = ⟨T*y, x⟩ for all x and y.
2
Banach-space adjoints replace inner products with dual pairings: (T′y′)(x) = y′(Tx).
3
Unbounded adjoints require domains: the defining identity is enforced only for x ∈ D(T).
4
Dense definition (D(T) dense in X) is the key assumption that makes the unbounded adjoint well defined and unique.
5
In both Banach and Hilbert settings, the adjoint comes with a maximal domain consistent with the defining identity.
6
Even when an adjoint exists, its domain can be too small (possibly only {0}), limiting how much information T* provides.

Highlights

Dense definition is what turns a “domain-restricted” adjoint identity into a well-defined, unambiguous operator.

Banach-space adjoints map continuous linear functionals to continuous linear functionals, using (T′y′)(x) = y′(Tx).

In Hilbert spaces, the unbounded adjoint T* is defined by requiring ⟨y, Tx⟩ = ⟨T*y, x⟩ for all x in D(T).

The adjoint’s existence doesn’t guarantee usefulness: its domain might be extremely small. 

Topics

Adjoint Operators
Unbounded Operators
Banach Space Duals
Hilbert Space Inner Products
Dense Domains