Some Construction of Generalized Frames with Adjointable Operators in Hilbert C∗−modules

Let T(t), t ≥ 0, be a C0-semigroup of linear operators acting in a Hilbert space H with norm ‖·‖. We prove that T(t) is uniformly bounded, i.e., ‖T(t)‖ ≤ M, t ≥ 0, if and only if the following condition is satisfied: $$\mathop {\sup }\limits_{t > 0} \frac{1}{t}\int\limits_0^t {\left\| {(T(s) + T^ * (s))x} \right\|^2 ds < \infty for all x \in H,} $$ , where T* is the adjoint operator.

Geophysical Inverse Theory and Regularization Problems

Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all A∈ A(X) and some B ∈ L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.

The purpose of this paper is to prove the following result.Let X be a complex Hilbert space, let L (X) be the algebra of all bounded linear operators on X and let A (X) ⊂ L (X) be a standard operator algebra, which is closed under the adjoint operation.Let T : A (X) → L (X) be a linear mapping satisfying the relation 3T (AA * A) = T (A)A * A + AT(A * )A + AA * T (A) for all A ∈ A (X).In this case T is of the form T (A) = λ A for all A ∈ A (X), where λ is some fixed complex number.

Some inequalities similar to Jensen inequalities for general Lebesgue integral are obtained. Applications for functions of self adjoint operators and functions of unitary operators on complex Hilbert spaces are provided as well.

Scattering theory for long range potentials

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

The first four sections of this chapter form its core and include classical topics such as bounded linear transformations, the open mapping theorem, the closed graph theorem, the uniform boundedness principle, and the Hahn-Banach theorem. The chapter includes a good number of applications of the four fundamental theorems of functional analysis. Sections 6.5 and 6.6 provide a good account of the properties of the spectrum and adjoint operators on Banach spaces. They may be largely bypassed, since the treatment of the corresponding topics for operators on Hilbert spaces in chapter 7 is self-contained. The section on weak topologies is more advanced and may be omitted if a brief introduction is the goal. The chapter is enriched by such topics as the best polynomial approximation, the Hilbert cube, Gelfand’s theorem, Schauder bases, complemented subspaces, and the Banach-Alaoglu theorem.

Let {fl(n) n Z} be a sequence of positive numbers with/3(0) 1 and sup{fl(n + 1)/fl(n) n Z} < .We then define the Hilbert space L2(fl) by L2(fl) L2(Z, f12).We denote f L2(fl) by f E= f(n)z with the summand f(n)z indicatin that f(n) is the n-th term of the sequence.Thus we have Ilfll :=-[f(n)12(fl(n))2.We also use z 1.The set {Z n Z} can be thought of as an othogonal basis for L2(/3).For f, g L2(/3) we define the formal product h fg by h .l (n)z where'(n)= E (k)g(n-k) k= if all the latter sums converge.Now let L(fl) {o L2(fl) of L2(/3) for all f LE(fl)}.Then for o L(fl) we can define the linear map M r LE(fl) --LE(fl) by Mr(f) of.If we let wn fl(n / 1)/fl(n), we have the following theorem.THEOREM (Shields, [3]).For o L(fl), M r is a bounded linear operator on LE(fl) and Mz is unitarily equivalent to the bilateral weighted shift T with weight sequence {Wn n Z}.Furthermore, under this unitary equiv- alence, {M r o L(fl)} corresponds to the commutant of T.One could also start with a bilateral weighted shift with a weight sequence of positive terms and then define {fl(n) n Z} by:In this paper when we say weighted shift, we are referring to a bilateral weighted shift with all terms of the weight sequence being positive.For the most part, we will be studying weighted shifts with periodic weight sequences.Theorem 2 provides some information about the adjoints of operators in a certain subalgebra of L(/3), and Theorem 3 is a spectral inclusion theorem for shifts with periodic weight sequences.

Today's quantum processors composed of fifty or more qubits have allowed us to enter a computational era where the output results are not easily simulatable on the world's biggest supercomputers. What we have not seen yet, however, is whether or not the complexity achievable with such quantum processors would ever be useful for any practical applications. In the quantum neural network approach, there has been an expectation that the Hilbert space can serve as a computational resource by providing a large feature space; however, it is not understood, nor illustrated in what way the Hilbert space provides that computational power. In this work we introduce a resource-efficient quantum neural network model with a focus on the role of the feature space the quantum processors can give. Then we apply it for classification tasks, showing a quantum advantage in terms of the physical resources needed to realize the neural network. To facilitate the Hilbert space as a large feature space, we utilize scale-free networks that can be generated in a discrete-time crystal model. The virtue of this approach is in both its theoretical simplicity and its practical applicability: our quantum neural network model is simple enough to identify the role of the Hilbert space as the feature space, and, furthermore, can be implemented with current technology. Our model does not require optimization of the quantum processor, and hence once the quantum processor has been set, it can be used for other different classification problems without any changes in the quantum processor itself.

Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δn}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δn}gn∈N is an inner *-higher derivation.

Let 𝒜 {\mathcal{A}} be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ {\mathcal{H}} which is closed under adjoint operation. Suppose that ϕ : 𝒜 → 𝒜 {\phi:\mathcal{A}\to\mathcal{A}} is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ {\ast} -derivation.

We present a general scheme for 3D geophysical inversion using a deep learning convolutional neural network that enables three-dimensional inversions of useful size to be solved on laptops and desktops. A priori constraints of maximum smoothness or compactness on model parameters used during conventional geophysical inversion are not necessary. In environments where a single stabilizing functional is not capable of adequately representing the subsurface variation (e.g., steel cased wells, buried pipes in smoothly varying geology), the method provides an attractive alternative. Projecting data to the Hilbert space of model parameters using the adjoint operator before training mitigates non uniqueness associated with the underdetermined nature of traditional 3D geophysical inverse problems. A field inversion example using magnetic data collected in Washington -on- Brazos state historic site to image buried pipe and other infrastructure is presented. While the training for a single inversion takes longer, the results show better resolution of the top of the structure and overall shape compared to conventional minimum structure inversion methods. For multiple inversions such as for systems with moving footprints, or time lapse surveys, neural networks can be substantially faster than traditional inversion as the network needs to be trained only once for several successive model predictions. The method can be applied to other geophysical data including seismic, electromagnetic and gravity, at various scales and resolution.

Composition operators induced by injective homomorphisms on infinite weighted graphs

We construct peculiar Hilbert spaces from counterexamples to the axiom of choice. We identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with such spaces. Here a self adjoint operator is intrinsically effective if and only if the Schrödinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.Mathematics Subject Classification: 03E35, 81P10, 03E25, 35Q40, 46B26, 47A60.