Open Mapping Theorem

The Open Mapping Theorem is a fundamental result in functional analysis that deals with the properties of linear operators between Banach spaces. It essentially states that a continuous linear surjective operator between two Banach spaces maps open sets to open sets. This has profound implications for the invertibility of such operators.

Statement of the Theorem

Let $X$ and $Y$ be Banach spaces, and let $T: X \to Y$ be a continuous linear operator. If $T$ is surjective (onto), then $T$ is an open mapping. This means that for any open set $U$ in $X$, the image $T(U)$ is an open set in $Y$.

Understanding "Open Mapping"

An open set in a topological space is a set where every point has a neighborhood that is entirely contained within the set. For a function (or operator) to be an open mapping, it must transform every open set in its domain into an open set in its codomain.

Why Banach Spaces are Crucial

The theorem relies heavily on the completeness property of Banach spaces. If either $X$ or $Y$ is not a Banach space (e.g., just a normed vector space), the theorem does not necessarily hold. Completeness ensures that certain iterative processes converge, which is essential for the proof.

Proof Outline (Conceptual)

The proof is typically involved and uses Baire's Category Theorem. Here's a simplified conceptual breakdown:

• Assume $T$ is a continuous linear surjective operator. We want to show that $T$ maps the open unit ball $B_X(0, 1)$ in $X$ to an open set in $Y$. If we can show this, then any open set $U$ can be represented as a union of open balls, and since $T$ maps these balls to open sets, $T(U)$ will also be open.
• Consider the sequence of closed balls $B_X(0, n)$. Since $T$ is surjective, the union of their images $T(\bigcup_{n=1}^{\infty} B_X(0, n)) = Y$.
• Baire's Category Theorem states that a complete metric space cannot be written as the countable union of nowhere dense closed sets. This implies that at least one of the sets $T(\overline{B_X(0, n)})$ must have a non-empty interior.
• Using properties of linear operators and the fact that $T$ is continuous, it can be shown that if $T(\overline{B_X(0, n)})$ has a non-empty interior, then $T(B_X(0, 1))$ must contain an open ball centered at the origin in $Y$.
• If $T(B_X(0, 1))$ contains an open ball, then $T$ maps open sets to open sets.

Consequences of the Open Mapping Theorem

One of the most important consequences is the Inverse Mapping Theorem.

Inverse Mapping Theorem

Let $X$ and $Y$ be Banach spaces, and let $T: X \to Y$ be a continuous linear operator. If $T$ is bijective (both injective and surjective), then its inverse $T^{-1}: Y \to X$ is also continuous.

Proof Sketch: If $T$ is bijective and continuous, it is surjective. By the Open Mapping Theorem, $T$ is an open map. This means $T$ maps open sets in $X$ to open sets in $Y$. Consider the identity map $I: X \to X$. $T^{-1} \circ T = I$. The inverse of an open map is a closed map, and a closed bijection between topological spaces has a continuous inverse. Since $T$ is an open map, $T^{-1}$ is continuous.

Closed Graph Theorem

The Closed Graph Theorem provides a criterion for a linear operator between Banach spaces to be continuous. It relates the continuity of an operator to the property of its graph being closed in the product space.

Understanding the Graph of an Operator

For a linear operator $T: X \to Y$, its graph, denoted by $G(T)$, is the subset of the product space $X \times Y$ defined as: $G(T) = \{ (x, T(x)) \in X \times Y \mid x \in X \}$ The graph $G(T)$ is equipped with the product topology of $X$ and $Y$.

Statement of the Theorem

Let $X$ and $Y$ be Banach spaces, and let $T: X \to Y$ be a linear operator. If $T$ is a closed operator, then $T$ is continuous.

An operator $T$ is called closed if its graph $G(T)$ is a closed subset of $X \times Y$ in the product topology. This means that if a sequence $\{(x_n, T(x_n))\}$ in the graph converges to some $(x, y) \in X \times Y$, then it must be that $y = T(x)$, and thus $(x, y)$ is in the graph $G(T)$.

Relationship to Continuity

Every continuous linear operator between Banach spaces is necessarily a closed operator. The Closed Graph Theorem states the converse: every closed linear operator between Banach spaces is continuous.

Why this is important: Often, it's easier to prove that an operator's graph is closed than to prove its continuity directly. The theorem allows us to deduce continuity from this closed graph property.

Proof Outline (Conceptual)

The proof of the Closed Graph Theorem relies on the Open Mapping Theorem.

• Consider the operator $T: X \to Y$, where $X$ and $Y$ are Banach spaces. Assume $T$ is closed.
• Define a new operator $\tilde{T}: X \to X \times Y$ by $\tilde{T}(x) = (x, T(x))$. The graph of $\tilde{T}$ is $G(\tilde{T}) = \{ (x, (x, T(x))) \mid x \in X \}$. This is not the graph of $T$.
• Instead, consider the operator $S: X \to Y$ such that its graph is $G(T)$. If $T$ is closed, then $G(T)$ is a closed subspace of $X \times Y$.
• Let's rephrase: Define a linear operator $L: X \to Y$ by $L(x) = T(x)$. If $T$ is closed, then the graph of $L$ is closed in $X \times Y$.
• Consider the product space $X \times Y$, which is a Banach space if $X$ and $Y$ are. The graph $G(T)$ is a subspace of $X \times Y$. If $T$ is closed, $G(T)$ is a closed subspace.
• Define a linear map $P_1: G(T) \to X$ by $P_1(x, T(x)) = x$. This map is clearly surjective.
• Define a linear map $P_2: G(T) \to Y$ by $P_2(x, T(x)) = T(x)$.
• The key insight is to view $G(T)$ as a Banach space under the norm inherited from $X \times Y$. The operator $P_1$ maps the Banach space $G(T)$ to the Banach space $X$. $P_1$ is continuous. If $P_1$ is surjective and $G(T)$ is complete, then by the Open Mapping Theorem, $P_1$ is an open map.
• The continuity of $T$ can then be derived from the properties of $P_1$ and $P_2$. Specifically, if $T$ is closed, then the map $(x, T(x)) \mapsto x$ from $G(T)$ to $X$ is a continuous bijection. By the Inverse Mapping Theorem, its inverse $(x) \mapsto (x, T(x))$ is also continuous. This implies that $T(x)$ must be continuous in $x$.

The Converse: Continuous Implies Closed

Let $T: X \to Y$ be a continuous linear operator between Banach spaces $X$ and $Y$. Let $\{(x_n, T(x_n))\}$ be a sequence in the graph $G(T)$ that converges to $(x, y) \in X \times Y$.

Since $T$ is continuous, $T(x_n) \to T(x)$ as $x_n \to x$.

The convergence in $X \times Y$ means $x_n \to x$ in $X$ and $T(x_n) \to y$ in $Y$.

By the uniqueness of limits in $Y$, we must have $y = T(x)$.

Thus, $(x, y) = (x, T(x))$, which means $(x, y) \in G(T)$. Therefore, $G(T)$ is closed.

Properties of Conjugate Operators

The concept of conjugate operators (also known as adjoint operators) is central to the study of linear operators, especially in Hilbert spaces. For an operator $T$ on a Hilbert space $H$, its conjugate operator $T^*$ is defined such that $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x, y \in H$. While the topic mentions conjugate operators, the properties are most richly explored in the context of Hilbert spaces. We will discuss properties relevant to general Banach spaces and then focus on Hilbert spaces where the concept is most standard.

Conjugate Operator in Banach Spaces

In a general Banach space $X$, the conjugate (or dual) operator $T': X' \to X'$ of a bounded linear operator $T: X \to X$ is defined using the dual space $X'$. For $f \in X'$, $T'f$ is an element of $X'$ defined by: $(T'f)(x) = f(Tx)$ for all $x \in X$.

Here, $X'$ denotes the topological dual space of $X$, which consists of all continuous linear functionals on $X$.

Properties of the Dual Operator $T'$

• Linearity: If $T$ is linear, then $T'$ is linear.
• Boundedness: If $T$ is a bounded operator, then $T'$ is also a bounded operator, and $\|T'\| = \|T\|$. This is a significant result.
• Composition: For operators $T, S: X \to X$, we have $(TS)' = S'T'$. The order is reversed, similar to matrix transposition.
• Identity: If $I$ is the identity operator on $X$, then $I' = I$, the identity operator on $X'$.
• Null Space and Range: The null space of $T'$ is related to the closure of the range of $T$. Specifically, $\ker(T') = (\text{ran}(T))^{\perp}$, where $(\cdot)^{\perp}$ denotes the annihilator. This is a version of the Fredholm alternative.
• Injectivity/Surjectivity: $T$ is injective if and only if $\ker(T') = \{0\}$ (the zero functional). $T$ is surjective if and only if $\text{ran}(T')$ is dense in $X'$.

Conjugate (Adjoint) Operator in Hilbert Spaces

In a Hilbert space $H$, the Riesz Representation Theorem allows us to identify $H$ with its dual space $H'$ via an inner product. For a bounded linear operator $T: H \to H$, its adjoint operator $T^*: H \to H$ is uniquely defined by the relation: $\langle Tx, y \rangle_H = \langle x, T^*y \rangle_H$ for all $x, y \in H$.

The adjoint $T^*$ is also a bounded linear operator.

Properties of the Adjoint Operator $T^*$ in Hilbert Spaces

• Linearity and Boundedness: $T^*$ is linear and bounded, and $\|T^*\| = \|T\|$.
• Second Adjoint: $(T^*)^* = T$. The adjoint operation is an involution.
• Composition: For operators $T, S: H \to H$, we have $(TS)^* = S^*T^*$. Again, the order is reversed.
• Identity: $I^* = I$.
• Null Space and Range:
  • $\ker(T^*) = (\text{ran}(T))^{\perp}$
  • $\ker(T) = (\text{ran}(T^*))^{\perp}$
  This is a powerful relationship. The null space of the adjoint is the orthogonal complement of the range of the original operator, and vice versa.
• Injectivity/Surjectivity:
  • $T$ is injective if and only if $\ker(T^*) = \{0\}$.
  • $T$ is surjective if and only if $\text{ran}(T^*)$ is dense in $H$. (In finite dimensions, surjective implies kernel is {0} and range is H).
• Self-Adjoint Operators: An operator $T$ is self-adjoint if $T = T^*$. These operators have very special properties, including real eigenvalues and orthogonal eigenvectors, forming the basis of spectral theory.
• Normal Operators: An operator $T$ is normal if $TT^* = T^*T$. Normal operators are generalizations of self-adjoint and unitary operators and also have a rich spectral theory.
• Unitary Operators: An operator $U$ is unitary if $U^*U = UU^* = I$. Unitary operators preserve the inner product and norm, representing isometries. They are essential for understanding symmetries and transformations in quantum mechanics.
• Closedness: If $T$ is a bounded operator, then $T^*$ is also a bounded operator. Bounded operators are continuous, and as we know from the Closed Graph Theorem, continuous operators have closed graphs. The adjoint of a bounded operator is also bounded, implying its graph is closed.

Example: Adjoint of a Matrix

Consider a matrix $A$ representing a linear operator on $\mathbb{C}^n$ (with the standard complex inner product). The adjoint operator $A^*$ corresponds to the conjugate transpose of the matrix $A$, often denoted as $A^{\dagger}$ or $A^H$. If $A = \begin{pmatrix} 1+i & 2 \\ 3 & 4-i \end{pmatrix}$, then $A^* = \begin{pmatrix} 1-i & 3 \\ 2 & 4+i \end{pmatrix}$. The property $\langle Ax, y \rangle = \langle x, A^*y \rangle$ holds for vectors $x, y \in \mathbb{C}^n$.

Example: Adjoint of an Integral Operator

Let $H = L^2([a, b])$ be the Hilbert space of square-integrable functions on the interval $[a, b]$. Consider the integral operator $T$ defined by: $(Tf)(x) = \int_a^b K(x, t) f(t) dt$ where $K(x, t)$ is the kernel. The adjoint operator $T^*$ is given by: $(T^*g)(t) = \int_a^b \overline{K(s, t)} g(s) ds$ This means the kernel of the adjoint operator is the complex conjugate of the transposed kernel of the original operator. If the kernel $K(x, t)$ is real and symmetric ($K(x, t) = K(t, x)$), then $T$ is self-adjoint ($T=T^*$).