This chapter introduces bounded and compact operators from a separable Hilbert space to itself, and the eigenvalues and singular values of compact operators. Pseudodifferential operators on the Euclidean plane are introduced. Technical results on double operator integration of bounded operators and estimates for product-convolution operators required for later chapters are derived or referenced.
In this book, we assume a graduate level knowledge of functional analysis. The introduction to operators and traces is brief; Chapter 2 in Volume I provides more detail. Some results from Volume I concerning traces on the weak trace class ideal of bounded operators are recalled in Theorems1.1.10 and1.1.13. We provide only the results, some without proof, required for applications of singular traces in Alain Connes’ noncommutative geometry studied in later chapters. Notation is also established.
1.1 Bounded operators on Hilbert space and traces
LetH be a separable complex infinite-dimensional Hilbert space. If ⟨·,·⟩ denotes the inner product onH with associated norm ‖·‖, then a linear operatorA:H→H is bounded if
‖A‖∞:=supη∈H,‖η‖=1‖Aη‖<∞.
Denote byL(H) the algebra of bounded linear operators ofH to itself. The rank of a bounded linear operatorA, denoted byrank(A), is the dimension of the rangeAH. The operatorA is compact if there is a sequence{Rk}k=0∞ of finite-rank operators onH such that
limk→∞‖A−Rk‖∞=0.
The set of compact linear operators withinL(H) is denoted byC0(H).
An operatorA∈L(H) is positive, denotedA≥0, if⟨Aη,η⟩≥0 for allη∈H. The adjoint ofA∈L(H) is the operatorA∗∈L(H) defined by⟨A∗η,ξ⟩=⟨η,Aξ⟩ for allη,ξ∈H. An operatorA∈L(H) is self-adjoint ifA is equal to its own adjointA∗. An operatorU∈L(H) is called unitary if
U∗U=UU∗=1,
whereU∗ is the adjoint ofU. Unitary operators are the isometries mapping the Hilbert spaceH to itself. Bounded operatorsA,B∈L(H) are unitarily equivalent if
A=U∗BU
for some unitary operatorU∈L(H).
Ifx∈l∞ is a complex-valued bounded sequence and{en}n=0∞ is an orthonormal basis ofH, thenx can be associated with the diagonal operatordiag(x)∈L(H) acting by
diag(x)η:=∑n=0∞x(n)⟨η,en⟩en,η∈H.
The diagonal operator depends on the orthonormal basis; diagonal operators for different orthonormal bases are unitarily equivalent. Many results for traces hold up to unitary equivalence, so the basis used to define the embeddingdiag ofl∞ inL(H) is specified only when necessary.
1.1.1 Singular values and ideals
A compact operatorA∈C0(H) has a sequence of singular values defined in the same way as the singular values of a matrix of complex numbers in linear algebra. Recall that the spectrum of a compact operator is a discrete set composed of eigenvalues, each associated with a finite-dimensional eigenspace of eigenvectors, and with 0 as the only limit point; see Chapter 2 in Volume I. The dimension of an eigenspace is called the multiplicity of the associated eigenvalue.
Definition 1.1.1.
An eigenvalue sequence ofA∈C0(H) is a sequence
λ(A)={λ(n,A)}n=0∞∈c0
of the eigenvalues ofA listed with multiplicity, with zeros appended ifA has only a finite number of eigenvalues, and such that the sequence|λ(n,A)|,n∈Z+, is nonincreasing.
A positive operator0≤A∈C0(H) has a unique eigenvalue sequence. For a general operatorA∈C0(H), given two eigenvalue sequencesλ(A)′ andλ(A) ofA, the operatorsdiag(λ(A)′) anddiag(λ(A)) are unitarily equivalent.
The absolute value|A| of a bounded operatorA∈L(H) can be obtained from the productA∗A ofA and its adjointA∗ by using the spectral theorem to take the square root of the positive operatorA∗A. The singular value sequence{μ(n,A)}n=0∞ of a compact operatorA∈C0(H) is defined by the eigenvalue sequence of|A|,
μ(n,A):=λ(n,|A|),n∈Z+.
The notion of the singular value sequence of a compact operator is extended to a bounded operator by the singular value function. The singular value function of a bounded operator on a separable Hilbert spaceH is discussed in Chapter 2 of Volume I.
Definition 1.1.2.
The singular value functionμ(A) ofA∈L(H) is defined by
μ(t,A):=inf{‖A−R‖∞:rank(R)≤t},t≥0.
Note thatμ(diag({μ(n,A)}n=0∞))=μ(A) for allA∈L(H) and thatμ(t,A),t>0, is a step function. The valuesμ(n,A),n∈Z+, will be referred to as the singular values ofA∈L(H). When it is clear that a sequence is referred to,μ(A) also denotes the singular values. WhenA∈C0(H) is compact, the singular values are the ordered eigenvalues of|A| as above. Whenx∈l∞, the sequence
μ(n,x):=μ(n,diag(x)),x∈l∞,n≥0,
is the decreasing rearrangement ofx.
Singular values characterize the ideal structure of the algebraL(H).
Definition 1.1.3.
A two-sided ideal of the algebraL(H) is a subspaceJ(H) such that
AB,BA∈J(H),A∈J(H),B∈L(H).
We discuss only two-sided ideals. The space of compact operators ofH to itself forms an ideal insideL(H).
Definition 1.1.4.
A linear subspaceJ ofl∞ is called a symmetric sequence space if
μ(n,y)≤μ(n,x),n≥0,
fory∈l∞ andx∈J implies thaty∈J.
John Williams Calkin proved that the singular value function provides a bijective correspondence between the ideals ofL(H) and the symmetric sequence spaces withinl∞.
Theorem 1.1.5 (Calkin correspondence).Let J be a symmetric sequence space. Then the subspaceJ(H)of...
