Problem 34 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

If \(\left(A, D_{A}\right)\) is a densely defined openator and \(D_{A}\) is dense in \(\mathcal{H}_{1}\) show that \(A \subseteq A^{* *}\).

Step-by-Step Solution

Answer

The operator \(A\) is a subspace of its double adjoint \(A^{**}\) because for each \(x\) in its domain, \(Ax\) is in the domain of \(A^{**}\) and \(Ax = A^{**}x\).

Step 1: Introduction to Elements and Spaces

Here, \(A\) is an operator and \(D_A\) is the domain of \(A\) which is densely defined in the Hilbert space \(\mathcal{H}_1\). This means that \(D_A\) is a subset of \(\mathcal{H}_1\) and that the closure of \(D_A\) is \(\mathcal{H}_1\). The problem statement is also saying that \(A\) is densely defined because its domain \(D_A\) is dense in \(\mathcal{H}_1\). The goal is to show that \(A \subseteq A^{**}\).

Step 2: Defining the Adjoint

The adjoint of an operator \(A\), denoted \(A^*\), is an operator such that for all \(x\) in \(D_A\) and \(y\) in the domain of \(A^*\), we have \(\langle Ax, y \rangle = \langle x, A^* y \rangle\). Here, \(\langle ., . \rangle\) denotes the inner product in the Hilbert space.

Step 3: Defining the Second Adjoint

The second adjoint of \(A\), denoted \(A^{**}\), is defined as the adjoint of \(A^*\). For all \(x\) in the domain of \(A^*\) and \(y\) in the domain of \(A^{**}\), we have \(\langle A^* x, y \rangle = \langle x, A^{**}y \rangle\).

Step 4: Showing Inclusion

To show that \(A \subseteq A^{**}\), we need to show that for each \(x\) in \(D_A\), \(Ax\) is also in the domain of \(A^{**}\) and \(Ax = A^{**}x\). This can be shown using the definitions of adjoints. For each \(x\) in \(D_A\) and each \(y\) in the domain of \(A^*\), since \(A \subseteq A^*\), we have \(\langle Ax, y \rangle = \langle x, A^*y \rangle = \langle x, A^{**}y \rangle\). Thus, we find that \(Ax\) is in the domain of \(A^{**}\) and \(Ax = A^{**}x\). Therefore, \(A \subseteq A^{**}\).

Key Concepts

Operator Theory

Operator theory is a branch of functional analysis that deals with operators on function spaces, such as Hilbert spaces and Banach spaces. Operators are like functions, which map elements from one space to another. They play a vital role in various mathematical and physical applications. In this context, operators are not merely number functions but abstract mappings that can act on infinitely dimensional spaces.

Key aspects of operator theory:

• Linear Operators: These operators satisfy linear conditions, which means they preserve vector addition and scalar multiplication.
• Bounded and Unbounded: Bounded operators have restrictions on their outputs, while unbounded ones do not. This distinction is important for mathematical stability.
• Spectrum and Eigenvalues: In operator theory, studying the spectrum (analogous to eigenvalues in matrices) provides deeper insights into an operator's properties.

Understanding operator theory provides the tools to study complex systems in quantum mechanics, electrical engineering, and more.

Adjoint Operator

The concept of an adjoint operator is crucial in operator theory, especially within the setting of Hilbert spaces. The adjoint of an operator, denoted as \(A^*\), is an operator that relates to the original operator \(A\) through the inner product. If you think of transposing a matrix, that idea extends to adjoint operators.

Key characteristics:

• Inner Product Relation: For operators \(A\) and \(A^*\), the adjoint relationship is given by \( \langle Ax, y \rangle = \langle x, A^*y \rangle \). This relationship is a cornerstone for defining self-adjoint or Hermitian operators.
• Importance: In applications, adjoint operators often describe how a system behaves under certain transformations or symmetries.
• Knowing the adjoint can lead to understanding the stability and responses of a system when subjected to inputs.

Understanding adjoints helps in solving problems like partial differential equations and quantum mechanics where self-adjoint operators represent observable quantities.

Dense Domain

In the realm of functional spaces like Hilbert spaces, the notion of a dense domain is significant. A domain is dense in a space if every element in that space can be approximated as closely as desired by elements from the domain. Think of it like trying to fill a room with tiny beads; you want them densely packed so that there are no gaps left.

Key elements:

• Mathematical Closeness: If a domain is dense, any point in the whole space can be reached by a limit of points from the domain.
• Implications in Analysis: With dense domains, many operator-associated problems become more tractable, as every function in the larger space can be handled by considering the domain.
• Integration in Problems: Dense domain conditions ensure that certain defined operators can operate meaningfully over the entire space, making them highly applicable in real-world scenarios.

Dense domains play a critical role in ensuring that operators defined in mathematical models are applicable and solve problems faithfully across entire spaces.

Inner Product Space

An inner product space is a vector space equipped with an additional structure called an inner product. This inner product allows for measuring angles and lengths, introducing geometric intuition and analytical convenience into abstract algebraic settings. Inner product spaces lay the foundational ground of understanding Hilbert spaces, essential for many areas of mathematics and physics.

Core features:

• Inner Product Definition: Typically denoted by \( \langle x, y \rangle \), the inner product of two vectors gives a scalar value. This value helps determine orthogonality (perpendicularity) and lengths.
• Geometric Interpretations: Thanks to inner products, we can talk about projections, angles, and distances in infinite-dimensional spaces just like in our 3D-visible world.
• Uses in Theory: In Hilbert spaces, inner products are used to define and explore concepts like orthonormal bases and projections, pivotal for expanding functions and signals.

Understanding inner product spaces enriches the study of mathematical fields such as quantum mechanics, where the angle between states can mean the probability of a state transition.