搜索结果：operators

We solve the following problems associated with Toeplitz operators $T_Φ$ on Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$. $(I)$ Given operator-valued bounded analytic functions $Γ, Ψ$ on $\mathbb{D}^n$, we completely characterize when the product $M_ΓM_Ψ^*$ becomes a Toeplitz operator by identifying tractable conditions on the functions. Furthermore, these conditions can be used to explicitly write the product into a sum of simple Toeplitz operators. $(II)$ We prove that partially isometric Toeplitz operators admit the following factorization: \[ T_Φ = M_Γ M_Ψ^*, \] where, $Γ, Ψ$ are operator-valued inner functions on $\mathbb{D}^n$. A few of the immediate consequences are: $(a)$ every partially isometric Toeplitz operator has a partially isometric symbol almost everywhere on $\mathbb{T}^n$ (distinguished boundary of $\mathbb{D}^n$), $(b)$ any partially isometric analytic Toeplitz operator is of the form $M_{ΓV^*}$, where $Γ$ is an operator-valued inner function and $V$ is an constant isometry. In connection with the result $(ii)$, we establish and use a crucial phenomenon: the range of partially isometric Toeplitz operat

We enumerate the complete and independent sets of operators at the next-to-leading order (NLO) in the Higgs effective field theory (HEFT), based on the Young tensor technique on the Lorentz, gauge and flavor structures. The operator-amplitude correspondence tells a type of operators forms the on-shell amplitude basis, and for operators involving in Nambu-Goldstone bosons, the amplitude basis is further reduced to the subspace satisfying the Adler's zero condition in the soft momentum limit. Different from dynamical field, the spurion should not enter into the Lorentz sector, instead it only plays a role of forming the $SU(2)$ invariant together with other dynamical fields. With these new treatments, for the first time we could obtain the 224 (7704) operators for one (three) generation fermions, 295 (11307) with right-handed neutrinos, and find there were 6 (9) terms of operators missing and many redundant operators can be removed in the effective theory without (with) right-handed neutrinos.

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-su

Given a self-adjoint operator $T$ on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set $\mathcal D(T)$ of all possible diagonals of $T$. For compact operators $T$, we give a complete characterization of diagonals modulo the kernel of $T$. That is, we characterize $\mathcal D(T)$ for the class of operators sharing the same nonzero eigenvalues (with multiplicities) as $T$. Moreover, we determine $\mathcal D(T)$ for a fixed compact operator $T$, modulo the kernel problem for positive compact operators with finite-dimensional kernel. Our results generalize a characterization of diagonals of trace class positive operators by Arveson and Kadison and diagonals of compact positive operators by Kaftal, Loreaux, and Weiss. The proof uses the technique of diagonal-to-diagonal results, which was pioneered in the earlier joint work of the authors with Siudeja.

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-Δ)$ and a second-order differential operator whose leading coefficients are not positive semi-definite.

In this paper we revisit the hypothesis needed to define the "paracomposition" operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by S. Alinhac in [3]. More precisely we do so in two directions. First we drop the diffeomorphism hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces. Thus we fully generalize Bony's classic paralinearasition theorem giving sharp estimates for composition in Sobolev and Zygmund spaces. In order to prove that the new class of operations benefits of symbolic calculus properties when composed by a paradifferential operator, we discuss the pull-back of pseudodifferential and paradifferential operators which then become Fourier Integral Operators. In this discussion we show that those Fourier Integral Operators obtained by pull-back are pseudodifferential or paradifferential operators if and only if they are pulled-back by a diffeomorphism that is a change of variable. We give a proof of the change of variables in paradifferential operators. Finally we study the cutoff defining paradifferential operators and it's stability by successive composition. It is known that the cutoff becom

According to Kim, Peris and Song, a continuous linear operator $T$ on a complex Banach space $X$ is called {\it numerically hypercyclic} if the numerical orbit $\{f(T^nx):n\in\N\}$ is dense in $\C$ for some $x\in X$ and $f\in X^*$ satisfying $\|x\|=\|f\|=f(x)=1$. They have characterized numerically hypercyclic weighted shifts and provided an example of a numerically hypercyclic operator on $\C^2$. We answer two questions of Kim, Peris and Song. Namely, we construct a numerically hypercyclic operator, whose square is not numerically hypercyclic as well as an operator which is not numerically hypercyclic but has two numerical orbits whose union is dense in $\C$. We characterize numerically hypercyclic operators on $\C^2$ as well as the operators similar to a numerically hypercyclic one and those operators whose conjugacy class consists entirely of numerically hypercyclic operators. We describe in spectral terms the operator norm closure of the set of numerically hypercyclic operators on a reflexive Banach space. Finally, we provide criteria for numeric hypercyclicity and decide upon the numerical hypercyclicity of operators from various classes.

To be able to solve operator equations numerically a discretization of those operators is necessary. In the Galerkin approach bases are used to achieve discretized versions of operators. In a more general set-up, frames can be used to sample the involved signal spaces and therefore those operators. Here we look at the redundant representation of operators resulting from a matrix representation using frames. We focus on injectivity, surjectivity and, in particular, invertibility of the involved operators and matrices. Furthermore we show sufficient conditions that the composition of matrices correspond to the composition of operators.

Bilocal light-ray operators which are Lorentz scalars, vectors or antisymmetric tensors, and which appear in various hard scattering QCD processes, are decomposed into operators of definite twist. These operators are harmonic tensor functions and their Taylor expansion consists of (traceless) local light-cone operators with span irreducible representations of the Lorentz group with definite spin j and common geometric twist (= dimension - spin). Some applications concerning the nonforward matrix elements of these operators and the generalization fo conformal light-cone operators of definite twist is considered. The group theoretical background of the method has been made clear.

Square of a posinormal operator is not necessarily posinormal$.$ But (i) powers of quasiposinormal operators are quasiposinormal and, under closed ranges assumption, powers of (ii) posinormal operators are posinormal, (iii) of operators that are both posinormal and coposinormal are posinormal and coposinormal, and (iv) of semi-Fredholm posinormal operators are posinormal.

We introduce and investigate $H^\infty$-functional calculus for commuting finite families of Ritt operators on Banach space $X$. We show that if either $X$ is a Banach lattice or $X$ or $X^*$ has property $(α)$, then a commuting $d$-tuple $(T_1,\ldots, T_d)$ of Ritt operators on $X$ has an $H^\infty$ joint functional calculus if and only if each $T_k$ admits an $H^\infty$ functional calculus. Next for $p\in(1,\infty)$, we characterize commuting $d$-tuple of Ritt operators on $L^p(Ω)$ which admit an $H^\infty$ joint functional calculus, by a joint dilation property. We also obtain a similar characterisation for operators acting on a UMD Banach space with property $(α)$. Then we study commuting $d$-tuples $(T_1,\ldots, T_d)$ of Ritt operators on Hilbert space. In particular we show that if $\Vert T_k\Vert\leq 1$ for every $k=1,\ldots,d$, then $(T_1,\ldots, T_d)$ satisfies a multivariable analogue of von Neumann's inequality. Further we show analogues of most of the above results for commuting finite families of sectorial operators.

The paper is devoted to operators given formally by the expression \begin{equation*} -\partial_x^2+\big(α-\frac14\big)x^{-2}. \end{equation*} This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real $α$, or closed operator for complex $α$, we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on $L^2({\mathbb R}_+)$, which we denote $H_{m,κ}$ and $H_0^ν$, with $m^2=α$, $-1<\Re(m)<1$, and where $κ,ν\in{\mathbb C}\cup\{\infty\}$ specify the boundary condition at $0$. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always $[0,\infty[$. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us

We construct operators for simulating the scattering of two hadrons with spin on the lattice. Three methods are shown to give the consistent operators for PN, PV, VN and NN scattering, where P, V and N denote pseudoscalar, vector and nucleon. Explicit expressions for operators are given for all irreducible representations at lowest two relative momenta. Each hadron has a good helicity in the first method. The hadrons are in a certain partial wave L with total spin S in the second method. These enable the physics interpretations of the operators obtained from the general projection method. The correct transformation properties of the operators in all three methods are proven. The total momentum of two hadrons is restricted to zero since parity is a good quantum number in this case.

We study mixed-state localization operators from the perspective of Werner's operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore, we provide a description of the Cohen class in terms of Werner's convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen's class.

This paper considers several approximate operators used in a particle method based on a Voronoi diagram. We introduce and study our approximate operators on gradient and Laplace operators. We derive error estimates for these approximate operators by applying our weight functions. The key idea of deriving our error estimates is to divide the integration region into a ring-shaped area and some areas. In the Appendix, we give an example application of the main results of this paper.

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem $dU(t) = AU(t) dt + B dW(t); U(0) = 0$ Here, $A$ is the generator of a strongly continuous semigroup of operators on a Banach space $E$, $B$ is a bounded linear operator from a separable Hilbert space $H$ into $E$, and $W$ is an $H$-cylindrical Brownian motion. When $A$ and $B$ are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family $ \{\sqrtλ R(λ, A)B : λ> 0\} $ is uniformly gamma-radonifying. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

Sequential effect systems are a class of effect system that exploits information about program order, rather than discarding it as traditional commutative effect systems do. This extra expressive power allows effect systems to reason about behavior over time, capturing properties such as atomicity, unstructured lock ownership, or even general safety properties. While we now understand the essential denotational (categorical) models fairly well, application of these ideas to real software is hampered by the sheer variety of source level control flow constructs in real languages. Denotational approaches are general enough to accommodate any particular control flow construct, but provide no guidance on the details, let alone applications. We address this new problem by appeal to a classic idea: macro-expression of commonly-used programming constructs in terms of control operators. We give an effect system for a subset of Racket's tagged delimited control operators, as a lifting of an effect system for a language without direct control operators. This gives the first account of sequential effects in the presence of general control operators. Using this system, we also re-derive the seq

We present covariant symmetry operators for the conformal wave equation in the (off-shell) Kerr-NUT-AdS spacetimes. These operators, that are constructed from the principal Killing-Yano tensor, its `symmetry descendants', and the curvature tensor, guarantee separability of the conformal wave equation in these spacetimes. We next discuss how these operators give rise to a full set of conformally invariant mutually commuting operators for the conformally rescaled spacetimes and underlie the $R$-separability of the conformal wave equation therein. Finally, by employing the WKB approximation we derive the associated Hamilton-Jacobi equation with a scalar curvature potential term and show its separability in the Kerr-NUT-AdS spacetimes.

Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is natural. However, it seems that the first results in this direction have been obtained only recently, to the best of our knowledge. Here we consider sectorial operators on scales of Hilbert spaces. We define a notion of convergence that generalises convergence of the resolvents in operator norm to the case when the operators act on different spaces and show that this kind of convergence is compatible with the functional calculus of the operator and moreover implies convergence of the spectrum. Finally, we present examples for which this convergence can be checked, including convergence of coefficients of parabolic problems. Convergence of a manifold (roughly speaking consisting of thin tubes) towards the manifold's skeleton graph plays a prominent role, being our main application.