Question

Let \(A\) be a bounded linear operator mapping a Banach space E into itself. Show that if \(A^{*}\) is completely continuous, then so is \(A\).

Short answer

If \( A^{*} \) is completely continuous, then \( A \) is also completely continuous by duality and operator properties.

Step-by-step solution

Understand the Definitions

A linear operator \( A \) from a Banach space \( E \) into itself is completely continuous if it maps weakly convergent sequences into strongly convergent sequences. A similar definition applies to \( A^{*} \), the adjoint operator acting on the dual space \( E^{*} \).

Use Properties of the Adjoints

If \( A^{*} \) is completely continuous, then for any weakly convergent sequence of functionals \( \{f_n\} \) in \( E^{*} \), \( A^{*}(f_n) \) converges strongly in \( E^{*} \). This property will be leveraged to demonstrate the complete continuity of \( A \).

Connect Through Duality

Consider a weakly convergent sequence \( \{x_n\} \) in \( E \). For each \( f \in E^{*} \), \( f(x_n) \to f(x) \). Thus, \( A^{*}f(x_n) \to A^{*}f(x) \) given \( A^{*} \) is completely continuous.

Construct the Argument for A

Since \( A^{*}(f_n)(x_n) \to A^{*}(f)(x) \) implies \( (A^{*}f)(Ax_n) \to (A^{*}f)(Ax) \), it follows that \( A \) is completely continuous, as \( Ax_n \to Ax \) for weakly convergent \( x_n \).

Conclude the Proof

By demonstrating that every weakly convergent sequence is sent to a strongly convergent sequence through \( A \), it can be concluded that \( A \) is completely continuous, assuming the complete continuity of its adjoint \( A^{*} \).

Key concepts

Banach Space

A Banach space is an essential concept in functional analysis, representing a complete normed vector space. To understand this better, let’s break down what it means. A vector space is said to be 'normed' when it is equipped with a function called a 'norm'. This norm assigns a length or size to each vector in the space. When we say it is 'complete,' it means that any Cauchy sequence of vectors (a sequence where the vectors get arbitrarily close to each other) has a limit within the space itself. Banach spaces are crucial because they provide a setting where mathematical limits behave very nicely, allowing for robust analysis of advanced concepts. Examples include spaces like the sequence space \( l^p \) and spaces of continuous functions. These spaces pave the way for deep theoretical insights and practical applications across mathematics and physics.

Bounded Linear Operator

A bounded linear operator is a fundamental concept used to understand transformations in spaces like Banach spaces. It refers to a linear transformation between normed vector spaces that satisfies a particular boundedness condition. Specifically, a linear operator \( A \) from one space \( E \) to another \( F \) is bounded if there exists a constant \( C \) such that for every vector \( x \) in \( E \), the inequality \( \|Ax\| \leq C\|x\| \) holds. This condition ensures the operator doesn't distort vectors too much, preserving their bounded nature.Bounded linear operators are important because they maintain nice properties under integration and limits and allow for the extension of finite-dimensional insights to infinite-dimensional settings, which is crucial in functional analysis and its applications.

Adjoint Operator

In the realm of functional analysis, an adjoint operator is an important extension in the study of operators between spaces. If you have a bounded linear operator \( A \) from Banach space \( E \) into \( F \), its adjoint \( A^* \) is an operator from the dual space \( F^* \) into \( E^* \). This becomes particularly interesting because of the way these operators interact with dual pairings.The adjoint operator is defined such that for all elements in \( F^* \) and \( E \), the relationship \( f(Ax) = (A^*f)(x) \) holds true. This property elegantly captures how functionals applied after and before transformation by \( A \) maintain consistency, allowing deeper insights into the behavior of \( A \) through its adjoint. The adjoint operator is pivotal when exploring concepts like complete continuity, as seen in duality arguments and spectral theory.

Weak Convergence

Weak convergence is a subtle concept in functional analysis, crucial when dealing with infinite-dimensional spaces. A sequence \( \{x_n\} \) in a Banach space \( E \) is said to converge weakly to a vector \( x \) if, for every continuous linear functional \( f \) in the dual space \( E^* \), it holds that \( f(x_n) \to f(x) \).Weak convergence is less stringent than strong convergence (where \( \|x_n - x\| \to 0 \)), allowing sequences to converge in a more generalized sense. This is significant in many applications as it provides a more refined tool for analyzing convergence, particularly when dealing with operators and functions in infinite-dimensional spaces. Understanding weak convergence helps in the exploration of properties like compactness and continuity, which are central in mathematical analysis.