Boundedness of Self-Adjoint Operators on Finite-Dimensional Hilbert Spaces

In the realm of linear algebra and functional analysis, a significant question arises regarding the boundedness of operators on a finite-dimensional Hilbert space. Specifically, we aim to prove that every self-adjoint operator on a finite-dimensional Hilbert space is bounded.

Introduction to Hilbert Spaces and Operators

A Hilbert space is a vector space equipped with an inner product that allows the definition of length and angle. This space is central in quantum mechanics and functional analysis. An operator on a Hilbert space is a function that maps one vector in the space to another. The concept of boundedness is crucial in this context, as it ensures that the operator does not map sequences of vectors to arbitrarily large values.

Self-Adjoint Operators

A self-adjoint operator on a Hilbert space is a linear operator that satisfies the condition (A A^*), where (A^*) denotes the adjoint of the operator (A). In simpler terms, for a self-adjoint operator on a finite-dimensional Hilbert space, there exists a basis in which the matrix representation of the operator is real and symmetric.

Boundedness in Finite-Dimensional Hilbert Spaces

In a finite-dimensional Hilbert space, every operator is inherently bounded, regardless of the norm or inner product defined on the space. This result stems from the fact that a finite-dimensional vector space is isomorphic to (mathbb{C}^n) (or (mathbb{R}^n) in the real case), where any infinite-dimensional phenomenon (like unboundedness) does not arise.

Proof of Boundedness for Self-Adjoint Operators

Step 1: Operator Representation

Consider an operator (A) on a finite-dimensional Hilbert space. By the finite-dimensional spectral theorem, (A) can be represented as a matrix (M) in some orthonormal basis, where (M) is Hermitian (i.e., (M M^*)). The matrix elements of (M) are the eigenvalues of (A).

Step 2: Boundedness of the Matrix

The matrix (M) is bounded because its norm (the maximum absolute value of its eigenvalues) is finite. This follows from the fact that in a finite-dimensional space, there is a maximum eigenvalue and a corresponding eigenvector. Any other vector in the space can be expressed as a linear combination of these eigenvectors, and the operator's action on the vector will be bounded by a multiple of this maximum eigenvalue.

Step 3: Operator Norm and Continuity

The operator norm of (A), denoted as (|A|), is defined as the maximum value of (|Ax|) for all unit vectors (x) in the Hilbert space. Given the finite dimensions and the boundedness of the matrix, the operator norm (|A|) is finite. This confirms that (A) is a bounded operator.

Implications and Applications

The boundedness of self-adjoint operators on finite-dimensional Hilbert spaces has profound implications in both mathematics and physics. In quantum mechanics, for instance, operators representing observables (like energy or position) must be bounded to ensure that the corresponding measurements are physically meaningful.

Conclusion

In summary, the proof that every self-adjoint operator on a finite-dimensional Hilbert space is bounded is grounded in the fundamental properties of finite-dimensional vector spaces and the spectral theorem. This result not only highlights the interplay between algebra and analysis but also underscores the importance of boundedness in the realm of operator theory.

Keywords

bounded operators self-adjoint operators Hilbert spaces

Additional Reading

For further exploration, consider reading about the spectral theorem for self-adjoint operators and the theory of bounded linear operators in Hilbert spaces.