Extract Although examples drive this book, we first provide a whirlwind survey of the general concepts of operator theory. We do not expect the student to master these topics now since they are covered in future chapters. A Hilbert space H is a complex vector space endowed with an inner product ⟨x,y⟩ that defines a norm ‖x‖=⟨x,x⟩ with respect to which H is (Cauchy) complete. The inner product on a Hilbert space satisfies the Cauchy–Schwarz inequality |⟨x,y⟩|⩽‖x‖‖y‖ for all x,y∈H. Examples of Hilbert spaces include Cn (complex Euclidean space), ℓ2 (the space of square-summable complex sequences), and L2[0,1] (the Lebesgue space of square-integrable, complex-valued functions on [0,1]). Vectors x,y in a Hilbert space H are orthogonal if ⟨x,y⟩=0. The dimension of a Hilbert space H is the cardinality of a maximal set of nonzero orthogonal vectors. This book is almost exclusively concerned with Hilbert spaces of countable dimension. Every such Hilbert space has an orthonormal basis (un)n=1∞, a (possibly finite) maximal orthogonal set such that ⟨um,un⟩=δmn for all m,n⩾1. With respect to an orthonormal basis (un)n=1∞, each x∈H enjoys a generalized Fourier expansion x=∑n=1∞⟨x,un⟩un that satisfies Parseval’s identity ‖x‖2=∑n=1∞|⟨x,un⟩|2. A subspace (a norm-closed linear submanifold) of H is itself a Hilbert space with the operations inherited from H. If (wn)n=1∞ is a (possibly finite) orthonormal basis for a subspace M of H and x∈H, then PMx=∑n=1∞⟨x,wn⟩wn belongs to M and satisfies ‖x−PMx‖⩽‖x−y‖ for every y∈M. In short, PMx is the unique closest vector to x in M. Furthermore, PM defines a linear transformation on H whose range is M. It is called the orthogonal projection of H onto M and it satisfies PM2=PM and ⟨PMx,y⟩=⟨x,PMy⟩ for all x,y∈H. Let H and K be Hilbert spaces. A linear transformation A:H→K is bounded if ‖A‖=sup{‖Ax‖K:‖x‖H=1} is finite. Let B(H,K) denote the set of bounded linear operators from H to K. We write B(H) for B(H,H). The quantity ‖A‖ is the norm of A. Since B(H) is closed under addition and scalar multiplication, it is a vector space. Furthermore, since ‖A+B‖⩽‖A‖+‖B‖ and ‖cA‖=|c|‖A‖ for all A,B∈B(H) and c∈C, it follows that B(H) is a normed vector space. Endowed with this norm, B(H) is complete and thus forms a Banach space. Moreover, the composition AB belongs to B(H) and ‖AB‖⩽‖A‖‖B‖ for all A,B∈B(H). Therefore, B(H) is a Banach algebra. For each A∈B(H,K), there is a unique A*∈B(K,H) such that ⟨Ax,y⟩K=⟨x,A*y⟩H for all x∈H and y∈K. The operator A* is the adjoint of A; it is the analogue of the conjugate transpose of a matrix. One can show that A↦A* is conjugate linear, that A**=A, ‖A‖=‖A*‖, and ‖A*A‖=‖A‖2. This additional structure upgrades B(H) from a Banach algebra to a C*-algebra. One can exploit adjoints to obtain information about the kernel and range of an operator. For most of the operators A∈B(H) covered in this book, we give the matrix representation [A]=[⟨Auj,ui⟩]i,j=1∞ with respect to an orthonormal basis (un)n=1∞ for H. This matrix representation [A] defines a bounded operator x↦[A]x on the Hilbert space ℓ2 of square summable sequences that is structurally identical to A. Schur’s theorem helps us determine which infinite matrices define bounded operators on ℓ2. Many of the operators covered in this book, such as the Cesàro operator, the Volterra operator, weighted shifts, Toeplitz operators, and Hankel operators, have fascinating structured-matrix representations. An important class of operators is the compact operators. These are the A∈B(H) such that (Axn)n=1∞ has a convergent subsequence whenever (xn)n=1∞ is a bounded sequence in H. Equivalently, an operator is compact if it takes each bounded set to one whose closure is compact. Each finite-rank operator is compact and every compact operator is the norm limit of finite-rank operators. The compact operators form a norm-closed, *-closed ideal within B(H). Some operators have a particularly close relationship with their adjoint. For example, the operator M on L2[0,1] defined by (Mf)(x)=xf(x) satisfies M*=M. Such operators are selfadjoint. If µ is a positive finite compactly supported Borel measure on C, then the operator N on L2(μ) defined by (Nf)(z)=zf(z) satisfies (N*f)(z)=z―f(z), and thus N*N=NN*. Such operators are normal. The operator (Uf)(eiθ)=eiθf(eiθ) on L2(T) satisfies U*U=UU*=I. Such operators are unitary. Unitary operators preserve the ambient structure of Hilbert spaces and can serve as a vehicle to relate A∈B(H) with B∈B(K). We say that A is unitarily equivalent to B if there is a unitary U∈B(H,K) such that UAU*=B. Unitary equivalence is often used to identify seemingly complicated operators with relatively simple ones.
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