Abstract
Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.
Highlights
Since its inception, the mathematical treatment of quantum theory has generated many problems in measure and integration theory, some of which are still being worked out
The eigenvalues λ ∈ σ(A) of A is replaced by the spectral decomposition T = σ(T ) λ dP (λ) with respect to the self-adjoint spectral measure P associated with the self-adjoint linear operator T
For a quantum system in a state ψ ∈ H, the conventional interpretation of quantum measurement suggests that the number kP (E)ψk2 is the probability that an observation of the quantity represented by the self-adjoint operator T has its value in the Borel set E ⊆ σ(T )
Summary
The mathematical treatment of quantum theory has generated many problems in measure and integration theory, some of which are still being worked out. For a quantum system in a state ψ ∈ H, the conventional interpretation of quantum measurement suggests that the number kP (E)ψk is the probability that an observation of the quantity represented by the self-adjoint operator T has its value in the Borel set E ⊆ σ(T ). The operator-valued spectral measure uniquely associated with a quantum observable is a fundamental concept in quantum theory. Another problem of integration theory arising from quantum physics is the Feynman-Kac formula: Z. BτH → H (T, h) 7−→ T h, T ∈ L(H), h ∈ H, uniquely defines a continuous linear map J : L(H)⊗ By this means, the variational properties of the spectral measure E play no role in the definition of the first integral in (2). Rt the closure of 0 X ⊗ dW in L2 (P ⊗ P ) as X runs over all adapted simple processes; see [14], Theorem
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