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The Herglotz–Nevanlinna theorem in 픻 is a very classical, useful theorem with myriad applications. Here a complete proof of its operator-valued version is given, which is then used to prove the spectral theorem for unitary operators. [ABSTRACT FROM AUTHOR]

In this paper, we study a new discrete tree and the resulting branching process, which we call the erlang weighted tree(EWT). The EWT appears as the local weak limit of a random graph model proposed in La and Kabkab, Internet Math. 11 (2015), no. 6, 528–554. In contrast to the local weak limit of well‐known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi‐type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, and so forth. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein–Rutman eigenvalue β0$$ {\beta}_0 $$ and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if β0≤1$$ {\beta}_0\le 1 $$. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral theory. More precisely, we study the spectral families and their corresponding operators on reflexive Banach spaces. Assume A is a well-bounded operator on reflexive Lebesgue spaces then the operator A is a scalar-type spectral operator. We establish that if a weak spectral family E (λ) is concentrated on [ a , b ] then there exists a linear well-bounded operator A ∈ L (X) on the reflexive Banach space X , such that 〈 A (x) , y * 〉 = b 〈 x , y * 〉 − ∫ [ a , b ] 〈 x , E (λ) y * 〉 d λ holds for all x ∈ X and y * ∈ X * . We prove that by assuming (Φ , X) is a functional calculus on the measurable space (Z , Σ) , then there are a semi-finite measure space (Ω , F , μ) and solitary operator U : X → L p (Ω , F , μ) , and an injective pointwise continuous *-homomorphism F : M (Z , Σ) → M (Ω , F) , such that Φ (f) = U M F f U − 1 holds for any operator of the multiplication M F f that multiplies by function f. [ABSTRACT FROM AUTHOR]

In this paper, we deal with non-selfadjoint operators with the compact resolvent. Having been inspired by the Lidskii idea involving a notion of convergence of a series on the root vectors of the operator in a weaker – Abel-Lidskii sense, we proceed constructing theory in the direction. The main concept of the paper is a generalization of the spectral theorem for a non-selfadjoint operator. In this way, we come to the definition of the operator function of an unbounded non-selfadjoint operator. As an application, we notice some approaches allowing us to principally broaden conditions imposed on the right-hand side of the evolution equation in the abstract Hilbert space.

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator. [ABSTRACT FROM AUTHOR]

In this article we investigate the spectral properties of the infinitesimal generator of an infinite system of master equations arising in the analysis of the approach to equilibrium in statistical mechanics. The system under consideration thus consists of infinitely many first-order differential equations governing the time evolution of probabilities susceptible of describing jumps between the eigenstates of a differential operator with a discrete point spectrum. The transition rates between eigenstates are chosen in such a way that the so-called detailed balance conditions are satisfied, so that for a large class of initial conditions the given system possesses a global solution which converges exponentially rapidly toward a time independent probability of Gibbs type. A particular feature and a challenge of the problem under consideration is that in the infinite-dimensional functional space where the initial-value problem is well posed, the infinitesimal generator is realized as a non normal and non dissipative compact operator, whose spectrum therefore does not exhibit a spectral gap around the zero eigenvalue. [ABSTRACT FROM AUTHOR]

Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collections can occur for many carefully chosen parameter values. The logistic map is such a system, as we show in a rigorous companion paper. To illustrate these very common phenomena, we compare the Lorenz system and the logistic map and we show how extremely similar their graph bifurcation diagrams are in some parameter ranges. Typically, bifurcation diagrams show how attractors change as a parameter is varied. We call ours "graph bifurcation diagrams" to reflect that not only attractors but also unstable periodic orbits and chaotic saddles can be shown. Only the most prominent ones can be shown. We argue that, as a parameter is varied in the Lorenz system, there are uncountably many parameter values for which there are infinitely many nodes, and infinitely many of the nodes N 1 , N 2 , N 3 , ... , N ∞ can be selected so that the graph has an edge from each node to every node with a node with a higher number. The final node N ∞ is an attractor. [ABSTRACT FROM AUTHOR]

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.

The replication of any European contingent claim by a static portfolio of calls and puts with strikes forming a continuum, formally proven by Carr and Madan [Towards a theory of volatility trading. In Volatility: New Estimation Techniques for Pricing Derivatives, edited by R.A. Jarrow, Vol. 29, pp. 417–427, 1998 (Risk books)], is part of the more general theory of integral equations. We use spectral decomposition techniques to show that exact payoff replication may be achieved with a discrete portfolio of special options. We discuss applications for fast pricing of vanilla options that may be suitable for large option books or high frequency option trading, and for model pricing when the characteristic function of the underlying asset price is known. [ABSTRACT FROM AUTHOR]

Projection spectral analysis is investigated and refined in this paper, in order to unify principal component analysis and independent component analysis. Singular value decomposition and spectral theorems are applied to nonsymmetric correlation or covariance matrices with multiplicities or singularities, where projections and nilpotents are obtained. Therefore, the suggested approach not only utilizes a sum‐product of orthogonal projection operators and real distinct eigenvalues for squared singular values, but also reduces the dimension of correlation or covariance if there are multiple zero eigenvalues. Moreover, incremental learning strategies of projection spectral analysis are also suggested to improve the performance.

A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators, here the functional calculus is not treated as a subordinate but as the central concept. Based on five simple axioms for a "measurable functional calculus", the theory of such calculi is developed in detail, including spectral theory, uniqueness results and construction principles. Finally, the functional calculus form of the spectral theorem is stated and proved, with some proof variants being discussed. [ABSTRACT FROM AUTHOR]

Model Order Reduction is used to vastly speed up simulations but it also introduces an error to the simulation results, which needs to be controlled. The a‐posteriori error estimator of Ruiner et al. for second‐order systems, which is based on the residual, has the advantage of having provable upper bounds and being usable independently of the reduction method. Nevertheless a bottleneck is found in the offline phase, making it unusable for larger models. We use the spectral theorem, power series expansions, monotonicity properties, and self‐tailored algorithms to largely speed up the offline phase by one polynomial order both in terms of computation time as well as storage complexity. All properties are proven rigorously. This eliminates the aforementioned bottleneck. Hence, the error estimator of Ruiner et al. can finally be used for large, linear, second‐order mechanical systems reduced by any model reduction method based on Petrov–Galerkin reduction. The examples show speedups of up to 28.000 and the ability to compute much larger systems with a fixed amount of memory. [ABSTRACT FROM AUTHOR]

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose (adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite-dimensional situation, is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson's normal form for bounded positive operators on infinite-dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states. [ABSTRACT FROM AUTHOR]

Projection spectral analysis is investigated and refined in this paper, in order to unify principal component analysis and independent component analysis. Singular value decomposition and spectral theorems are applied to nonsymmetric correlation or covariance matrices with multiplicities or singularities, where projections and nilpotents are obtained. Therefore, the suggested approach not only utilizes a sum‐product of orthogonal projection operators and real distinct eigenvalues for squared singular values, but also reduces the dimension of correlation or covariance if there are multiple zero eigenvalues. Moreover, incremental learning strategies of projection spectral analysis are also suggested to improve the performance. [ABSTRACT FROM AUTHOR]

In the tensor space $${{\mathrm {Sym}}}^d \mathbb {R}^2$$ of binary forms we study the best rank k approximation problem. The critical points of the best rank 1 approximation problem are the eigenvectors and it is known that they span a hyperplane. We prove that the critical points of the best rank k approximation problem lie in the same hyperplane. As a consequence, every binary form may be written as linear combination of its critical rank 1 tensors, which extends the Spectral Theorem from quadratic forms to binary forms of any degree. In the same vein, also the best rank k approximation may be written as a linear combination of the critical rank 1 tensors, which extends the Eckart-Young theorem from matrices to binary forms. [ABSTRACT FROM AUTHOR]