Identity Operator Research Articles

Smiley Theorem for Spectral Operators Whose Radical Part Is Locally Nilpotent

Generalizing bicommutant theorem to the higher-order commutator case is very useful for representation theory of Lie algebras, which plays an important role in symmetry analysis. In this paper, we mainly prove that for any spectral operator A on a complex Hilbert space whose radical part is locally nilpotent, if a bounded operator B lies in the k-centralizer of every bounded linear operator in the l-centralizer of A, where k and l are two arbitrary positive integers satisfying l⩾k, then B must belong to the von Neumann algebra generated by A and the identity operator. This result generalizes a matrix commutator theorem proved by M. F. Smiley. To this aim, Smiley operators are defined and an example of a non-spectral Smiley operator is given by the unilateral shift, indicating that Smiley-type theorems might also hold for general spectral operators.

Homogeneous Banach Spaces as Banach Convolution Modules over M(G)

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p<∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers.

On simplicity of Lie algebras of compact operators: A direct approach

We investigate an algebraic variant of the Wojtyński problem on the simplicity of Lie algebras of compact operators on a separable infinite-dimensional complex Hilbert space. We prove the non-simplicity of Lie algebras of compact operators under a mild softness condition using the notion of soft-edged operator ideals introduced by Kaftal and Weiss. We believe our study offers a new perspective on the investigation of the simplicity of Lie algebras by relating it to the study of operator ideals.

Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fr\'echet derivative of $F(t,u)$ is H\"{o}lder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\|F'_u(t,w) - F'_u(t,v)\|\le \alpha(t)\|v - w\|^{q}$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v} = F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if $\sup_{t\ge 0}\int_0^t \alpha(\xi)\|U(t,\xi)\|\, d\xi<\infty$ and $\sup_{t\ge 0}\|U(t)\|<\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi)$ is the solution to the equation $\frac{d}{dt}{U}(t,\xi) = F'_u(t,0)U(t,\xi)$, $t\ge \xi$, $U(\xi,\xi)=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for $\lim_{t\to\infty}u(t) = 0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.

Subsymmetric bases have the factorization property

Let $(e_j)_{j=1}^\infty $ denote a Schauder basis for a Banach space $X$, and let $(e_j^*)_{j=1}^\infty $ denote the biorthogonal functionals. We say that $(e_j^*)_{j=1}^\infty $ has the factorization property if the identity operator $I_{X^*}$ on $X^

The Lipschitz bounded approximation property for operator ideals

In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.

Parseval Wavelet Frames on Riemannian Manifold

We construct Parseval wavelet frames in $$L^2(M)$$ for a general Riemannian manifold M and we show the existence of wavelet unconditional frames in $$L^p(M)$$ for $$1< p <\infty $$ . This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $$L^2(M)$$ , which was recently proven by Bownik et al. (Potential Anal 54:41–94, 2021). We also show a characterization of Triebel–Lizorkin $${\mathbf {F}}_{p,q}^s(M)$$ and Besov $${\mathbf {B}}_{p,q}^s(M)$$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on $${\mathbf {F}}_{p,q}^s(M)$$ and $${\mathbf {B}}_{p,q}^s(M)$$ spaces on manifolds M with bounded geometry.

Closed ideals in the algebra of compact-by-approximable operators

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra AX:=K(X)/A(X) on Banach spaces X failing the approximation property. The examples include the following: (i) if X has cotype 2, Y has type 2, AX≠{0} and AY≠{0}, then AX⊕Y has at least 2 closed ideals, (ii) there are closed subspaces X⊂ℓp for 4<p<∞ and X⊂c0 such that AX contains a non-trivial closed ideal, (iii) there is a Banach space Z such that AZ contains an uncountable lattice of closed ideals having the reverse order structure of the power set of the natural numbers. Some of our results involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators X→Y, where X⊂ℓp and Y⊂ℓq are closed subspaces for p≠q.

Embedding weighted Hardy spaces into Lebesgue spaces and generalized area operators

In this paper, we consider the weighted Hardy space induced by an weight ω. We characterize the positive Borel measures μ such that the identity operator is bounded when As an application, we obtain necessary and sufficient conditions for the boundedness of the generalized area operator from to

Aggregation of Weak Fuzzy Norms

Aggregation is a mathematical process consisting in the fusion of a set of values into a unique one and representing them in some sense. Aggregation functions have demonstrated to be very important in many problems related to the fusion of information. This has resulted in the extended use of these functions not only to combine a family of numbers but also a family of certain mathematical structures such as metrics or norms, in the classical context, or indistinguishability operators or fuzzy metrics in the fuzzy context. In this paper, we study and characterize the functions through which we can obtain a single weak fuzzy (quasi-)norm from an arbitrary family of weak fuzzy (quasi-)norms in two different senses: when each weak fuzzy (quasi-)norm is defined on a possibly different vector space or when all of them are defined on the same vector space. We will show that, contrary to the crisp case, weak fuzzy (quasi-)norm aggregation functions are equivalent to fuzzy (quasi-)metric aggregation functions.

Perturbative quantization of Einstein-Hilbert gravity embedded in a higher derivative model

In a perturbative approach Einstein-Hilbert gravity is quantized about a flat background. In order to render the model power counting renormalizable, higher order curvature terms are added to the action. They serve as Pauli-Villars type regulators and require an expansion in the number of fields in addition to the standard expansion in the number of loops. Renormalization is then performed within the BPHZL scheme, which provides the action principle to construct the Slavnov-Taylor identity and invariant differential operators. The final physical state space of the Einstein-Hilbert theory is realized via the quartet mechanism of Kugo and Ojima. Renormalization group and Callan-Symanzik equation are derived for the Green functions and, formally, also for the $S$-matrix.

More about operator order preserving

It is well known that increasing functions do not preserve operator order in general; nor do decreasing functions reverse operator order. However, operator monotone increasing or operator monotone decreasing functions do. In this article, we employ a convex approach to discuss operator order preserving or conversing functions. As an easy consequence of more general results, we find nonnegative constants γ and ψ such that A≤B implies f(B)≤f(A)+γ1ℋ andf(A)≤f(B)+ψ1ℋ, for the self-adjoint operators A, B on a Hilbert space ℋ with identity operator 1ℋ and for the convex function f whose domain contains the spectra of both A and B. The connection of these results to the existing literature will be discussed and the significance will be emphasized by some examples.

Lipschitz integral operators represented by vector measures

&lt;p&gt;In this paper we introduce the concept of Lipschitz Pietsch-p-integral &lt;br /&gt;mappings, (1≤p≤∞), between a metric space and a Banach space. We represent these mappings by an integral with respect to a vector&lt;br /&gt;measure defined on a suitable compact Hausdorff space, obtaining in this way a rich factorization theory through the classical Banach spaces C(K), L_p(μ,K) and L_∞(μ,K). Also we show that this type of operators fits in the theory of composition Banach Lipschitz operator ideals. For p=∞, we characterize the Lipschitz Pietsch-∞-integral mappings by a factorization schema through a weakly compact operator. Finally, the relationship between these mappings and some well known Lipschitz operators is given.&lt;/p&gt;

The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains

Using the spectral theory on the S-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form T=∑ℓ=13eℓaℓ(x)∂xℓ,x=(x1,x2,x3)∈Ω¯,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T=\\sum _{\\ell =1}^3e_\\ell a_\\ell (x)\\partial _{x_\\ell }, \\ \\ \\ x=(x_1,x_2,x_3)\\in \\overline{\\Omega }, \\end{aligned}$$\\end{document}where, Omega can be either a bounded or an unbounded domain in mathbb {R}^3 whose boundary partial Omega is considered suitably regular, overline{Omega } is the closure of Omega and e_ell , for ell =1,2,3 are the imaginary units of the quaternions mathbb {H}. The operators T_ell :=a_ell (x)partial _{x_ell }, for ell =1,2,3, are called the components of T and a_1, a_2, a_3: overline{Omega } subset mathbb {R}^3rightarrow mathbb {R} are the coefficients of T. In this paper we study the generation of the fractional powers of T, denoted by P_{alpha }(T) for alpha in (0,1), when the operators T_ell , for ell =1,2,3 do not commute among themselves. To define the fractional powers P_{alpha }(T) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T. In this paper we consider two different boundary conditions. If Omega is unbounded we consider Dirichlet boundary conditions. If Omega is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator sum _{ell =1}^3a_ell ^2(x)n_ell (x) partial _{x_ell }+a(x)I, for xin partial Omega , where I is the identity operator, a:partial Omega rightarrow mathbb {R} is a given function and n=(n_1,n_2,n_3) is the outward unit normal vector to partial Omega . The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type sum _{ell =1}^3a_ell (x)n_ell (x) partial _{x_ell }+b(x)I, xin partial Omega . For this reason we also discuss the conditions on the coefficients a_1, a_2, a_3: overline{Omega } subset mathbb {R}^3rightarrow mathbb {R} of T and on the coefficient b:partial Omega rightarrow mathbb {R} so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

Laplace transformers in norm ideals of compact operators

Laplace transformers in norm ideals of compact operators

Convolutional proximal neural networks and Plug-and-Play algorithms

In this paper, we introduce convolutional proximal neural networks (cPNNs), which are by construction averaged operators. For filters with full length, we propose a stochastic gradient descent algorithm on a submanifold of the Stiefel manifold to train cPNNs. In case of filters with limited length, we design algorithms for minimizing functionals that approximate the orthogonality constraints imposed on the operators by penalizing the least squares distance to the identity operator. Then, we investigate how scaled cPNNs with a prescribed Lipschitz constant can be used for denoising signals and images, where the achieved quality depends on the Lipschitz constant. Finally, we apply cPNN based denoisers within a Plug-and-Play framework and provide convergence results for the corresponding PnP forward-backward splitting algorithm based on an oracle construction.

Characterization of symmetrically $$\Delta $$-normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

Characterization of symmetrically $$\Delta $$-normed operator ideals which are interpolation spaces between Schatten–von Neumann ideals

New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

Abstract A new class of operators, larger than ∗ \ast -finite operators, named generalized ∗ \ast -finite operators and noted by Gℱ ∗ ( ℋ ) {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ ∗ ( ℋ ) = { ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) : ∥ T A − B T ∗ − λ I ∥ ≥ ∣ λ ∣ , ∀ λ ∈ C , ∀ T ∈ ℬ ( ℋ ) } . {{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.

Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

AbstractWe use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo‐differential operators, on a smooth manifold Ω with boundary , in the context of the non‐harmonic analysis of boundary value problems introduced in [29] in terms of a model operator . For symbols in the Hörmander class , we provide a non‐harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo‐differential operator is a compact operator in , or a Riesz operator in in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non‐harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations.

On Gauge Invariance of the Bosonic Measure in Chiral Gauge Theories

Gauge invariance of the measure associated with the gauge field is usually taken for granted, in a general gauge theory. We furnish a proof of this invariance, within Fujikawa’s approach. To stress the importance of this fact, we briefly review gauge anomaly cancellation as a consequence of gauge invariance of the bosonic measure and compare this cancellation to usual results from algebraic renormalization, showing that there are no potential inconsistencies. Then, using a path integral argument, we show that a possible Jacobian for the gauge transformation has to be the identity operator, in the physical Hilbert space. We extend the argument to the complete Hilbert space by a direct calculation.