Question

Let \(X, Y\) be Banach spaces and \(T \in \mathcal{B}(X, Y)\). If \(T\) is an isomorphism into \(Y\), is \(T^{*}\) necessarily an isomorphism into \(X^{*}\) ? Hint: No, embed \(\mathbf{R}\) into \(\mathbf{R}^{2}\).

Short answer

No, \( T^* \) is not necessarily an isomorphism if \( T \) is one.

Step-by-step solution

Understand the Problem

We need to determine if the adjoint operator (denoted as \( T^* \)) is necessarily an isomorphism if \( T \) is an isomorphism.

Define the Terms

Given \(X\) and \(Y\) are Banach spaces, an operator \( T \) is an isomorphism if it is bijective (both injective and surjective) and bounded with a bounded inverse.

Examine the Adjoint Operator

The adjoint operator \( T^* \) maps \( Y^* \) to \( X^* \), where \( Y^* \) and \( X^* \) are the dual spaces of \(Y\) and \(X\), respectively.

Use the Hint (Embedding)

Consider an embedding example: let \( T \) embed \( \mathbf{R} \) into \( \mathbf{R}^2 \). For instance, define \( T: \mathbf{R} \rightarrow \mathbf{R}^2 \) by \( T(x) = (x, 0) \).

Determine the Adjoints

For \( T(x) = (x, 0) \), the adjoint \( T^* \) maps functionals from \( \mathbf{R}^2 \) to \( \mathbf{R} \). However, \( T^* \) is not surjective because there exist functionals in \( \mathbf{R}^2 \) that cannot be represented solely via \( (x, 0) \).

Conclude the Requirement

Since the adjoint \( T^* \) is not surjective, \( T^* \) cannot be an isomorphism into \( \mathbf{R}^* \). Therefore, \( T^* \) is not necessarily an isomorphism even though \( T \) is.

Key concepts

Banach spaces

Banach spaces form an essential concept in functional analysis, representing a complete normed vector space. This means that any Cauchy sequence in a Banach space converges within that space. Key aspects to understand Banach spaces better include:

• Completeness: Every Cauchy sequence converges to a limit in the space.
• Norm: A function that assigns a strictly positive length or size to all vectors in the space except for the zero vector.
• Examples: Classic examples include the spaces of continuous functions, square-integrable functions, and p-normed sequences.

Banach spaces provide a generalized context where many theorems from finite dimensional spaces still hold true and play a groundbreaking role in the development of analysis.

Isomorphism

In the context of Banach spaces, an isomorphism is a linear operator that creates a bijective, bounded, and invertible mapping between two spaces. The primary characteristics of an isomorphism are:

• Bijective: The operator is both injective (one-to-one) and surjective (onto).
• Bounded: The operator's image under a norm is proportional to its preimage's norm, ensuring that it behaves well under limits.
• Inverse: There exists an inverse operator which is also bounded.

These properties make isomorphisms particularly powerful in functional analysis, enabling us to transfer problems between different Banach spaces while preserving structural properties.

Dual spaces

The dual space of a Banach space X, denoted as X*, is the space of all continuous linear functionals on X. Understanding dual spaces provides deeper insight into functional analysis. Important aspects include:

• Functionals: These are mappings from the original space X to the field of scalars, typically the real numbers or complex numbers.
• Continuity: Functionals in the dual space are continuous, meaning small changes in input result in small changes in output.
• Representation: Elements of the dual space can often be represented or characterized via specific elements in the original space.

Dual spaces allow us to extend concepts of orthogonality and projection from finite-dimensional spaces to infinite-dimensional contexts.

Embedding

An embedding is a type of function that maps one structure into another, typically with the intention of preserving certain properties of the original structure. When discussing Banach spaces, an embedding might take the form of a linear operator. Key points to consider are:

• Injective Mapping: An embedding must map distinct elements to distinct elements, maintaining the injective property.
• Structure Preservation: Essential properties such as vector operations and norms are preserved under the embedding.
• Example: Embedding \(\textbf{R}\) into \(\textbf{R}^2\) through \((x) \rightarrow (x, 0)\).

Embeddings are useful in demonstrating how smaller spaces can be viewed within larger contexts, often simplifying analyses and proofs.

Adjoint operators

The adjoint operator, denoted as T*, maps the dual space of Y back to the dual space of X when T is an operator from X to Y. Adjoint operators have significant theoretical implications, especially in the study of bounded operators. Core characteristics include:

• Definition: Given an operator T from X to Y, the adjoint operator T* works in reverse, mapping Y* to X*.
• Properties: The adjoint retains a relationship \(\forall y^* \in Y^*, T^*(y^*)(x) = y^*(T(x))\) for \- all \(\forall x \in X\).
• Non-surjectivity: Using embedding examples like \(\textbf{R} \rightarrow \textbf{R}^2\) reveals that the adjoint operator need not be surjective, and hence not an isomorphism.

Understanding adjoint operators is crucial for studying operator theory in more complex functional spaces contexts.