Tensor Product Of Operators: Domain And Graph Norm

Let's dive into the fascinating world of tensor products of operators, graph norms, and their domains. This is a crucial area in operator theory, especially when dealing with unbounded operators on Hilbert spaces. We'll explore how these concepts intertwine and what implications they have for mathematical analysis.

Defining the Hilbert Spaces and Unbounded Operators

Operator theory often begins with a solid foundation in Hilbert spaces. Consider two Hilbert spaces, ${ H_1 }$ and ${ H_2 }$. A Hilbert space is a complete inner product space, which means it’s a vector space equipped with an inner product that allows us to define notions like length and angle, and it’s complete in the sense that every Cauchy sequence converges within the space. These spaces are fundamental in quantum mechanics, signal processing, and many other areas of mathematics and physics. Now, let’s introduce an unbounded closed densely defined operator ${ A }$ on ${ H_1 }$. This operator ${ A }$ is not defined on the entire Hilbert space ${ H_1 }$, but only on a dense subspace ${ D(A) }$, which we call the domain of ${ A }$. The term “densely defined” means that the domain ${ D(A) }$ is “close” to the entire space ${ H_1 }$, in the sense that every vector in ${ H_1 }$ can be approximated arbitrarily closely by vectors in ${ D(A) }$. Furthermore, ${ A }$ is closed, which means that if we have a sequence ${ x_n }$ in ${ D(A) }$ that converges to some ${ x }$ in ${ H_1 }$, and ${ Ax_n }$ converges to some ${ y }$ in ${ H_1 }$, then ${ x }$ must be in ${ D(A) }$ and ${ Ax = y }$. This property is crucial for ensuring that the operator behaves well under limits.

Graph Norm: A Hilbert Subspace

The graph norm plays a vital role in understanding the properties of unbounded operators. We can equip the domain ${ D(A) }$ with the graph norm, denoted by ${ \| \cdot \|_A }$, which is defined as

${ \|x\|_A = \sqrt{\|x\|^2 + \|Ax\|^2}, \quad x \in D(A). }$

Here, ${ \| \cdot \| }$ represents the usual norm on the Hilbert space ${ H_1 }$. The graph norm essentially measures both the size of a vector and the size of its image under the operator ${ A }$. When ${ D(A) }$ is equipped with this graph norm, it becomes a Hilbert subspace of ${ H_1 }$. This means that ${ D(A) }$ is a Hilbert space in its own right, and it is also a subspace of ${ H_1 }$. The completeness of ${ D(A) }$ under the graph norm is a direct consequence of the fact that ${ A }$ is a closed operator. This Hilbert subspace structure allows us to apply Hilbert space techniques to study the properties of ${ A }$.

Tensor Products of Operators

Now, let's move on to the tensor product of operators. Consider another unbounded closed densely defined operator ${ B }$ on ${ H_2 }$, with its domain ${ D(B) }$ and graph norm ${ \| \cdot \|_B }$. The tensor product of ${ A }$ and ${ B }$, denoted by ${ A \otimes B }$, is an operator that acts on the tensor product of the Hilbert spaces ${ H_1 \otimes H_2 }$. Defining this operator and understanding its domain requires careful consideration.

Defining the Tensor Product Operator

The tensor product operator ${ A \otimes B }$ is initially defined on the algebraic tensor product ${ D(A) \otimes D(B) }$, which consists of finite linear combinations of elementary tensors ${ x \otimes y }$, where ${ x \in D(A) }$ and ${ y \in D(B) }$. The action of ${ A \otimes B }$ on an elementary tensor is given by

${ (A \otimes B)(x \otimes y) = Ax \otimes By. }$

This definition is then extended linearly to the entire algebraic tensor product ${ D(A) \otimes D(B) }$. However, since ${ A }$ and ${ B }$ are unbounded, we need to be careful about defining the domain of ${ A \otimes B }$ as a closed operator on ${ H_1 \otimes H_2 }$.

Determining the Domain of the Tensor Product

The crucial question is: what is the appropriate domain for the tensor product operator ${ A \otimes B }$? We want to find a domain such that ${ A \otimes B }$ remains a closed operator. A common choice for the domain of ${ A \otimes B }$ is the completion of ${ D(A) \otimes D(B) }$ with respect to a suitable norm. One such norm is the graph norm associated with ${ A \otimes B }$, which is defined similarly to the single operator case.

Graph Norm for Tensor Products

To define the graph norm for ${ A \otimes B }$, we first need to understand how ${ A \otimes B }$ acts on elements in its domain. The graph norm ${ \| \cdot \|_{A \otimes B} }$ on ${ D(A) \otimes D(B) }$ is given by

${ \|z\|_{A \otimes B}^2 = \|z\|^2 + \|(A \otimes B)z\|^2, }$

where ${ z \in D(A) \otimes D(B) }$, and ${ \| \cdot \| }$ is the norm on the Hilbert space ${ H_1 \otimes H_2 }$. The completion of ${ D(A) \otimes D(B) }$ with respect to this graph norm gives us a Hilbert space, which we can denote as ${ D(A \otimes B) }$. This space is the domain of the closure of the operator ${ A \otimes B }$.

Properties and Implications

Understanding the domain of ${ A \otimes B }$ and its properties is essential for several reasons. First, it allows us to rigorously define and work with tensor product operators, which are crucial in quantum mechanics for describing composite systems. Second, the graph norm provides a way to control the behavior of the operator, ensuring that it remains well-behaved under limits. Finally, the Hilbert space structure of ${ D(A \otimes B) }$ allows us to apply powerful tools from functional analysis to study the properties of ${ A \otimes B }$.

Sobolev Spaces and Tensor Products

Sobolev spaces are closely related to the domains of unbounded operators, especially in the context of partial differential equations. A Sobolev space ${ W^{k,2}(\Omega) }$ consists of functions defined on a domain ${ \Omega }$ in ${ \mathbb{R}^n }$ that have weak derivatives up to order ${ k }$ in ${ L^2(\Omega) }$. These spaces are Hilbert spaces when equipped with an appropriate norm.

Sobolev Spaces as Domains

In many cases, the domain of an unbounded operator arising from a differential operator can be identified with a Sobolev space. For example, consider the Laplacian operator ${ -\Delta }$ on ${ L^2(\Omega) }$ with appropriate boundary conditions. The domain of the Friedrichs extension of ${ -\Delta }$ is often a Sobolev space ${ H^2(\Omega) \cap H_0^1(\Omega) }$, where ${ H^2(\Omega) }$ consists of functions with second-order weak derivatives in ${ L^2(\Omega) }$, and ${ H_0^1(\Omega) }$ consists of functions in ${ H^1(\Omega) }$ that vanish on the boundary of ${ \Omega }$.

Tensor Products of Sobolev Spaces

When dealing with tensor products, the relationship between Sobolev spaces and domains becomes even more interesting. If we have two operators ${ A }$ and ${ B }$ whose domains are Sobolev spaces, say ${ D(A) = H^k(\Omega_1) }$ and ${ D(B) = H^l(\Omega_2) }$, then the domain of their tensor product ${ A \otimes B }$ is related to the tensor product of these Sobolev spaces. In general, the tensor product of Sobolev spaces ${ H^k(\Omega_1) \otimes H^l(\Omega_2) }$ is not necessarily equal to another standard Sobolev space. However, it is often a subspace of a Sobolev space on the product domain ${ \Omega_1 \times \Omega_2 }$.

Implications for PDEs

This connection between Sobolev spaces and tensor products has significant implications for the study of partial differential equations (PDEs). For instance, when solving PDEs on product domains, the solutions often lie in tensor product Sobolev spaces. Understanding the properties of these spaces is crucial for analyzing the regularity and behavior of solutions to these PDEs.

Practical Implications and Examples

Let's look at some practical implications and examples to solidify our understanding. Suppose ${ H_1 = H_2 = L^2([0,1]) }$, and consider the operator ${ A = B = -\frac{d^2}{dx^2} }$ with Dirichlet boundary conditions, i.e., ${ u(0) = u(1) = 0 }$. The domain of ${ A }$ and ${ B }$ is then ${ D(A) = D(B) = H^2([0,1]) \cap H_0^1([0,1]) }$.

Example: Heat Equation

Now, consider the tensor product operator ${ A \otimes B }$ acting on ${ L^2([0,1]^2) }$. This operator appears naturally when studying the heat equation in two spatial dimensions. The domain of ${ A \otimes B }$ is related to the space of functions on the square ${ [0,1]^2 }$ that satisfy certain boundary conditions. Specifically, the functions must vanish on the boundary of the square and have sufficient regularity so that the operator ${ A \otimes B }$ can be applied.

Applications in Quantum Mechanics

In quantum mechanics, tensor products of operators are used to describe composite systems. For example, if we have two particles described by Hilbert spaces ${ H_1 }$ and ${ H_2 }$, the composite system is described by the tensor product Hilbert space ${ H_1 \otimes H_2 }$. Operators on ${ H_1 }$ and ${ H_2 }$ that describe the individual particles can be combined using the tensor product to describe interactions between the particles.

Numerical Analysis

From a numerical analysis perspective, understanding the domain of tensor product operators is crucial for developing efficient numerical schemes. When discretizing PDEs on tensor product domains, one often uses tensor product bases, such as tensor product finite elements or spectral methods. The choice of basis and the regularity of the solution in the appropriate domain directly affect the accuracy and convergence of the numerical method.

Conclusion

In summary, the domain of the tensor product of operators and the graph norm are fundamental concepts in operator theory and functional analysis. These concepts play a crucial role in defining and analyzing unbounded operators, understanding Sobolev spaces, and solving partial differential equations. By carefully considering the domains of operators and equipping them with appropriate norms, we can gain deeper insights into the behavior of these operators and their applications in various fields of mathematics and physics. Understanding these abstract concepts allows us to tackle real-world problems in quantum mechanics, numerical analysis, and PDE theory, providing a solid foundation for further research and development.