Bounded Linear Operators

Abstract

The first section gives several characterizations of bounded linear operators and proves that a symmetric operator whose domain is the whole Hilbert space is actually bounded (Hellinger-Toeplitz theorem). Several concrete examples of bounded linear operators in Hilbert spaces are discussed in the second section. In Section 3 the vector space $\mathcal{L}(\mathcal{H},\mathcal{K})$ of all bounded linear operators from a Hilbert space $\mathcal{H}$ into a Hilbert space $\mathcal{K}$ over the same field is introduced and its basic properties are investigated. It turns out that the map $A \mapsto A^*$ which assigns the adjoint to a given element $A \in \mathcal{L}(\mathcal{H},\mathcal{K})$ defines an involution on $\mathcal{L}(\mathcal{H},\mathcal{K})$ which is isometric. It is also shown that the normed vector space $\mathcal{L}(\mathcal{H},\mathcal{K})$ is complete. In the next section the case of $\mathcal{L}(\mathcal{H},\mathcal{K})$ with $\mathcal{K}=\mathcal{H}$ is addressed. As usual we use then the notation ${\mathcal{B}(\mathcal{H})} = \mathcal{L}(\mathcal{H},\mathcal{H})$ . In this case the composition of operators is available and allows to define naturally a product in ${\mathcal{B}(\mathcal{H})}$ which gives this space the structure of a C$^*$ -algebra with unit. Hence one can do calculus in ${\mathcal{B}(\mathcal{H})}$ . In particular, given a power series $\sum_i a_i x^i$ with radius of convergence $r>0$ we can form $\sum_i a_i A^i$ and find that this is a well-defined element in ${\mathcal{B}(\mathcal{H})}$ whenever $\sum_i a_i A^i$ . This is investigated in Section 5. In particular we know what $\,\textrm{e}\,^A$ for $A \in{\mathcal{B}(\mathcal{H})}$ means. Another important implication is that we know to calculate the square root $\sqrt{A}$ of a positive operator A in ${\mathcal{B}(\mathcal{H})}$ by using the power series expansion of the function $\sqrt{1-z}$ (square root lemma). As an immediate consequence we prove the polar decomposition of operators in ${\mathcal{B}(\mathcal{H})}$ .