Understanding Bounded Linear Operators with Adjoint in Hilbert Spaces

Introduction

In the realm of operator theory and functional analysis, particularly within Hilbert spaces, the study of bounded linear operators is fundamental. One aspect that often perplexes students and practitioners is the relationship between bounded linear operators that admit adjoints and their surjectivity. This article delves into a specific scenario: whether every bounded linear operator that admits an adjoint is necessarily surjective in Hilbert spaces. Through the exploration of orthogonal projection operators, we will uncover the nuances of this question.

Bounded Linear Operators and Adjoint Operators

In a Hilbert space (H), a bounded linear operator (T: H to H) is a linear map that is also bounded, meaning there exists a constant (C > 0) such that for all (x in H), (|Tx| leq C|x|). The concept of an adjoint operator, denoted by (T^*), plays a crucial role in functional analysis. For any bounded linear operator (T), (T^*) is defined such that (langle Tx, y rangle langle x, T^*y rangle) for all (x, y in H).

Non-Surjective Bounded Linear Operators with Adjoint

To address the initial question, let us consider a concrete example: an orthogonal projection operator (P) onto a proper subspace (P) of (H). This operator is self-adjoint, meaning (P P^*), and it is a projection, satisfying (P^2 P). Importantly, (P) is not surjective because its range (the subspace onto which it projects) is a proper subspace of (H).

The proper in "proper subspace" means the subspace is neither the zero subspace nor the whole space (H). For instance, if (H L^2(mathbb{R})), the space of square-integrable functions on the real line, an orthogonal projection could be onto the subspace of functions that are zero outside a fixed interval, say ([-1, 1]).

Properties of the Orthogonal Projection Operator

The orthogonal projection operator (P) has several salient properties:

It is idempotent: (P^2 P). It is self-adjoint: (P P^*). It maps every vector in (H) onto a vector in the subspace onto which it is projecting. It is not surjective, as it maps (H) into the subspace but not necessarily onto the entire space.

These properties illustrate that it is indeed possible for a bounded linear operator that admits an adjoint to not be surjective. The operator (P) serves as a counterexample to the claim that every such operator must be surjective.

General Implications and Broader Context

The example of the orthogonal projection operator (P) onto a proper subspace highlights an important distinction between bounded linear operators that admit adjoints and those that are surjective. Surjectivity, in the context of Hilbert spaces, refers to the property that the image of the operator is the entire space. In contrast, the adjoint operator provides a powerful tool for analysis but does not inherently guarantee surjectivity.

For a bounded linear operator to be surjective, it must map the entire domain space onto the range space, which is not a necessary condition for the existence of an adjoint.

Conclusion

In conclusion, the study of bounded linear operators in Hilbert spaces reveals that not every operator that admits an adjoint is necessarily surjective. This is exemplified by the orthogonal projection operator (P), which is self-adjoint and idempotent but is not surjective, mapping (H) into a proper subspace. The existence of an adjoint operator is a different property, and it does not automatically imply surjectivity. Understanding these nuances is crucial for a deeper appreciation of operator theory and its applications in various areas of mathematics and physics.

Keywords: bounded linear operator, adjoint operator, Hilbert space, orthogonal projection, surjective.

References: [1] Banach, S. (1932). Theorie des operations lineaires. Monografie Matematyczne, Warszawa.

Further Reading:

Berberian, S. K. (1974). The Theory of Vector Lattices. Hilger. Zygmund, A. (2003). . Cambridge University Press.