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 \({\mathcal{B}(\mathcal{H})}\) .