Question

For any pair of hermitian operators \(A\) and \(B\) on a Hilbert space \(\mathcal{H}\), define \(A \leq B\) iff \((u \mid A u) \leq(u \mid B u)\) for all \(u \in \mathcal{H}\). Show that this is a partial order on the set of hermatian operators pay particular attention to the symmetry property, \(A \leq B\) and \(B \leq A\) umplies \(A=B\). (a) For multiplication operators on \(L^{2}(X)\) show that \(A_{0} \leq A_{B}\). Iff \(\alpha(x) \leq \beta(x)\) a.e. on \(X\). (b) For projection operators show that the definition given here reduces to that given in the text, \(P_{N} \leq P_{N}\) iff \(M \subseteq N\)

Short answer

The binary relation defined between hermitian operators is a partial order, satisfying reflexivity, antisymmetry, and transitivity. The antisymmetry was proved by considering positive and negative differences between the operators. For multiplication operators, it was shown that \(A_{0}\) is less than or equal to \(A_{B}\) if function \(\alpha\) is everywhere less or equal to \(\beta\). For projection operators, if \(P_{M}\) is less or equal to \(P{N}\), then subspace \(M\) must be a subset of subspace \(N\).

Step-by-step solution

Proving antisymmetry

Let's assume that \(A \leq B\) and \(B \leq A\). That means \((u|Au) \leq (u|Bu)\) and \((u|Bu) \leq (u|Au)\) for all \(u \in \mathcal{H}\). Then, \((u|Au)-(u|Bu) \leq 0\) and \((u|Bu)-(u|Au) \leq 0\) for all \(u\). This implies \((u|(B-A)u) = 0\) for all \(u\), therefore, any eigenvalue of \((B-A)\) must be zero. If the only eigenvalue is zero, then \(B-A = 0\) or \(A = B\)

Multiplication Operators

Within the context of multiplication operators on \(L^{2}(X)\), let \(A_{0}\) and \(A_{B}\) be given by multiplication by functions \(\alpha, \beta: X → \mathbb{R}\). If \(A_{0} \leq A_{B}\), for any \(y ∈ L^2(X)\), we have \((y| A_0 y) \leq (y|A_B y)\). Integral over \(X\) of \(|\alpha(x)y(x)|^2 \leq \) integral over \(X\) of \(|\beta(x)y(x)|^2\). Dividing by \(\| y \|_2^2\) and using Fatou's Lemma, we pointwise obtain \(\alpha(x) \leq \beta(x)\), a.e on \(X\).

Projection Operators

Let's assume that \(P_{M} \leq P_{N}\), \((u|P_M u) \leq (u|P_N u)\) for all \(u \in \mathcal{H}\). Since \(P_M\) and \(P_N\) are projection operators, their outputs lie in the subspaces \(M\) and \(N\) respectively. We know that \(\|P_N u\|^2 \geq \|P_M u\|^2\) for all \(u\). This inequality holds because the norm of a projection of \(u\) onto a subspace is less than or equal to the norm of \(u\) itself. But since the inequality holds for all \(u\), this implies that the subspace \(M\) must be contained within \(N\) or \(M \subseteq N\)

Key concepts

Hermitian Operators

Hermitian operators are a special class of operators frequently encountered in quantum mechanics and linear algebra. They are defined on a complex Hilbert space and possess distinct properties:

- **Self-Adjointness**: An operator \( A \) is Hermitian if it equals its own adjoint, meaning \( A = A^* \). In simpler terms, the matrix representing the operator is equal to its conjugate transpose.
- **Real Eigenvalues**: All eigenvalues of a Hermitian operator are real. This is crucial for quantum mechanics, where observable quantities correspond to real numbers.
- **Orthogonal Eigenvectors**: Eigenvectors corresponding to distinct eigenvalues of a Hermitian operator are orthogonal.

Understanding Hermitian operators is vital as they assure meaningful physical predictions, as only real numbers are observed in nature.

Partial Order

Partial order is a relation that extends the comparison of elements in a set. It has several key properties that define it:

- **Reflexivity**: Every element is comparable to itself. That means for Hermitian operators \( A \) on \( \mathcal{H} \), \( A \leq A \) for all \( A \) .
- **Antisymmetry**: If \( A \leq B \) and \( B \leq A \), then \( A = B \). This property is assured in the context of Hermitian operators via the eigenvalue condition showing \((u | (B-A)u) = 0\).
- **Transitivity**: If \( A \leq B \) and \( B \leq C \), then \( A \leq C \).

In operators like mathematical matrices or function spaces, partial order allows for a structured way to understand relationships without the need for all elements to be directly comparable.

Multiplication Operators

Multiplication operators are a straightforward yet essential concept in functional analysis, especially useful for working with function spaces like \( L^2(X) \).

- A multiplication operator \( M \) on \( L^2(X) \) is defined by multiplying a function by another fixed function. For instance, if \( M \) multiplies each function in \( L^2(X) \) by \( \alpha(x) \), we write \( M_y = \alpha(x) \cdot y(x) \) for all \( y \in L^2(X) \).
- The partial order \( A_0 \leq A_B \) translates to \( \alpha(x) \leq \beta(x) \) almost everywhere on \( X \). This comes from comparing the integrals of the squares of these functions, ensuring non-negative values.

This facilitates understanding functions' behavior across the domain \( X \) and lays groundwork for more complex mappings and transformations.

Projection Operators

Projection operators are a type of linear operator mainly used to project vectors onto subspaces. They are vital in decomposition techniques and various optimization problems.

- **Definition**: A projection operator \( P \) satisfies \( P^2 = P \). It effectively "compresses" input vectors into a subspace, ensuring the result remains within it.
- **Subspace Containment**: For projections \( P_M \) and \( P_N \), \( P_M \leq P_N \) when the subspace corresponding to \( P_M \) is contained in that of \( P_N \), or mathematically, \( M \subseteq N \).

Projection operators reveal how one can simplify complex vector spaces by focusing on smaller, more manageable subspaces, while retaining important structural properties.