Question

Let \(P, Q\) be projections in a Banach space \(X .\) Show that the following are equivalent: (i) \(P(X) \subset Q(X)\) and \(P^{*}\left(X^{*}\right) \subset Q^{*}\left(X^{*}\right)\) (ii) \(P Q=Q P=P\). (iii) \(P(X) \subset Q(X)\) and \(\operatorname{Ker}(Q) \subset \operatorname{Ker}(P)\).

Short answer

The statements (i), (ii), and (iii) are equivalent as shown by mutual implication.

Step-by-step solution

Given Statements

We are given that we need to show the equivalence of three statements involving projections in a Banach space and their duals.

Show (i) implies (ii)

Assume that (i) holds: \(P(X) \subset Q(X)\) and \(P^{*}\big(X^{*}\big) \subset Q^{*}\big(X^{*}\big)\).1. Let \(x \in X\). Then \(P(x) = Q(y)\) for some \(y \in X\) since \(P(X) \subset Q(X)\).2. Apply \(Q\) to both sides: \(Q(P(x)) = Q(Q(y)) = Q(y) = P(x)\). Hence, \(P(x) \in \operatorname{Im}(Q)\), so \(P(x) = Q(P(x))\).3. Similarly, since \(P^{*}\big(X^{*}\big) \subset Q^{*}\big(X^{*}\big)\), we get that for any \(\varphi \in X^*\), \(P^{*}(\varphi) = Q^{*}(\psi)\) for some \(\psi \in X^{*}\). Thus, \(P^{*}(\varphi) = Q^{*}(P^{*}(\varphi))\). 4. This shows that \(P = QP\) and hence \(PQ = QP = P\).

Show (ii) implies (iii)

Assume (ii): \(P Q = Q P = P\).1. Let \(x \in P(X)\). Then \(P(x) = x\), so \(Q(x) = Q(P(x)) = P(x) = x\), implying \(P(X) \subset Q(X)\).2. Now consider \(x \in \operatorname{Ker}(Q)\), i.e., \(Q(x) = 0\). Since \(P = QP\), we have \(P(x) = Q(P(x)) = Q(0) = 0\). This shows that \operatorname{Ker}(Q) \subset \operatorname{Ker}(P)\.

Show (iii) implies (i)

Assume (iii): \(P(X) \subset Q(X)\) and \operatorname{Ker}(Q) \subset \operatorname{Ker}(P)\.1. Let \(\varphi \in P^{*}(X^{*})\). Then there exists some \(\psi \in X^{*}\) such that \(\varphi = P^{*}(\psi)\). Since \(P(X) \subset Q(X)\), \(P(x) = Q(y)\) for some \(y \in X\), which implies \( = <\psi, P(x)> = <\psi, Q(y)> = \) and thus \(P^{*}(\psi) \in Q^{*}(X^{*})\).2. Similarly, any \(x \in \operatorname{Ker}(Q)\) implies \(Q(x) = 0\) which implies \(P(x) \= 0\) showing \operatorname{Ker}(Q) \subset \operatorname{Ker}(P)\.3. Therefore, \(P^{*}(X^{*}) \subset Q^{*}(X^{*})\).

Key concepts

Banach space

A Banach space is a complete normed vector space. This means that every Cauchy sequence in the space converges within the space. Banach spaces are foundational in functional analysis, providing a setting for rigorous study of continuous linear operators, series, and more.
For example, if you have a sequence of functions that gets closer and closer together as the sequence progresses (a Cauchy sequence), in a Banach space, this sequence will have a limit within the same space.

• Completeness: All Cauchy sequences converge.
• Normed: There is a function that assigns a length to vectors.

Understanding Banach spaces helps us analyze projections and operators effectively.

Projections

Projections are linear operators on a Banach space that, when applied twice, yield the same result as when applied once. In other words, for a projection operator P, we have P² = P.
Projections can break a space into two parts: the image (Im) and the kernel (Ker). For example, if P is a projection, then the image Im(P) is the set of all vectors that P maps to itself (i.e., P(x) = x).
Key aspects are:

• Idempotence: P² = P.
• Decomposition: Splits the space into image and kernel.

This is useful when analyzing conditions like P(X) ⊆ Q(X).

Dual space

The dual space of a Banach space X, denoted X*, consists of all continuous linear functionals on X. Essentially, these are functions that take a vector from X and return a scalar, maintaining linearity and continuity.
For any vector x in X and a functional φ in X*, we write φ(x) to represent the scalar output.
Important properties:

• Continuity: Maps converging sequences to converging sequences of scalars.
• Linearity: φ(ax + by) = aφ(x) + bφ(y) for all scalars a, b.

Understanding how projections interact with the dual space, as in P*(X*) ⊆ Q*(X*), is crucial in advanced topics.

Kernels

The kernel (Ker) of a linear operator is the set of vectors that map to zero. For an operator Q, Ker(Q) consists of all x in X such that Q(x) = 0.
Kernels provide insight into the null spaces of operators. For projections P and Q, the relationship Ker(Q) ⊆ Ker(P) implies that every vector annihilated by Q is also annihilated by P.
Key points to note:

• Null space: Set of vectors mapped to zero.
• Subspaces: Kernels are always linear subspaces.

Kernels help in understanding the structure of spaces when split by projections.

Image inclusion

Image inclusion refers to the relationship between the images (or ranges) of two operators. For projections P and Q, P(X) ⊆ Q(X) means that every vector in the image of P is also in the image of Q.
This concept is significant when determining equivalences between various conditions involving projections. For instance:

• Subset relationship: Im(P) is a subset of Im(Q).
• Overlap of ranges: All vectors in P(X) are covered by Q(X).

Understanding image inclusion is critical when proving statements like P = QP or ensuring that P(X) ⊆ Q(X) holds true.