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Let $ X_1, \ldots, X_n,Y $ X 1 , ... , X n , Y be classes of Banach spaces-valued sequences. An n-linear operator A between Banach spaces belongs to the ideal of $ (X_1, \ldots, X_n;Y) $ (X 1 , ... , X n ; Y) -summing multilinear operators if $ (A(x_j^1, \ldots, x_j^n))_{j=1}^\infty $ (A (x j 1 , ... , x j n)) j = 1 ∞ belongs to Y whenever $ (x_j^k)_{j= 1}^\infty $ (x j k) j = 1 ∞ belongs to $ X_k, k = 1, \ldots, n $ X k , k = 1 , ... , n. In this paper we develop techniques to generate non trivial symmetric ideals of this type. Illustrative examples and additional applications of the techniques are provided. [ABSTRACT FROM AUTHOR]

Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a density theorem for operator-valued frames via square-integrable representations restricted to closed subgroups of locally compact groups, which is a natural extension of the density theorem in classical Gabor analysis. More precisely, it is proved that for such an operator-valued frame, the index subgroup is co-compact if and only if the generator is a Hilbert–Schmidt operator. Then we present some applications of this density theorem, and in particular establish necessary and sufficient conditions for the existence of such operator-valued frames with Hilbert–Schmidt generators. We also introduce the concept of wavelet transform for Hilbert–Schmidt operators, and use it to prove that if the representation space is infinite-dimensional, then the system indexed by the entire group is Bessel system but not a frame for the space of all Hilbert–Schmidt operators on the representation space. [ABSTRACT FROM AUTHOR]

AbstractIn this work, we give some p-numerical radius inequalities for operators as well as for commutators of operators. As applications, when p = ∞, we improve some earlier numerical radius inequalities for operators. Also, we refine the triangle inequalities for the numerical radius and the operator norm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of λ -limited sets and λ -L-sets in a Banach space X and its dual X ∗ respectively, using the vector valued sequence spaces λ w ∗ (X ∗) and λ w (X) . We find characterizations for these sets in terms of absolutely λ -summing operators and investigate the relationship between λ -compact sets and λ -limited sets, with a particular focus on the crucial role played by a norm iteration property. We also consider λ -limited operators and show that this class is an operator ideal containing the ideal of λ -compact operators for a suitably restricted λ . Furthermore, we define a generalized Gelfand-Philips property for Banach spaces corresponding to an abstract sequence space. [ABSTRACT FROM AUTHOR]

In this paper, we prove restriction theorems for the Fourier–Laguerre transform and establish Strichartz estimates for the Schrödinger propagator e - i t L α for the Laguerre operator L α = - Δ - ∑ j = 1 n ( 2 α j + 1 x j ∂ ∂ x j ) + | x | 2 4 , α = (α 1 , α 2 , … , α n) ∈ (- 1 2 , ∞) n on R + n involving systems of orthonormal functions. The proof is based on a combination of some known dispersive estimate and the argument in Nakamura [Trans Am Math Soc 373(2), 1455–1476 (2020)] on torus. As an application, we obtain the global well-posedness for the nonlinear Laguerre–Hartree equation in Schatten space. [ABSTRACT FROM AUTHOR]

In this paper, the boundedness and compactness of generalized integration operators T g n , k between different analytic tent spaces in the unit disc are completely characterized. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce the Laguerre–Bessel $ L^2_\alpha $ L α 2 -multiplier operators and to give some results related to these operators as Plancherel's, Calderón's reproducing formulas and Heisenberg's, Donoho–Stark's uncertainty principles. Next, using the theory of reproducing kernels we give best estimates and an integral representation of the extremal functions related to this operators. [ABSTRACT FROM AUTHOR]

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R 3 subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L 2 ln (L) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region L Λ ⊂ R 3 as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln (L) factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂ Λ , is not solely on its area but on the "surface perpendicular to the direction of the magnetic field". We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L 2 of the relevant trace for piecewise C 1 , α smooth surfaces ∂ Λ . [ABSTRACT FROM AUTHOR]

Let Φ , Ψ be symmetrically norming (s.n.) functions, and L [ μ ] Δ A , B X = def L [ μ ] (Δ A , B ) X = def ∫ R + e - t A X e - t B d μ (t) denotes the Laplace transformer generated by the generalized derivation where μ is a Borel probability measure on If both pairs consist of mutually commuting accretive operators, such that both C - A and D - B are accretive and for some , then L [ μ ] Δ A ∗ , A (I) - L [ μ ] Δ C ∗ , C (I) ⩾ 0 , L [ μ ] Δ B , B ∗ (I) - L [ μ ] Δ D , D ∗ (I) ⩾ 0 and | | C ∗ + C - A ∗ - A (L [ μ ] Δ A , B X - L [ μ ] Δ C , D X) D + D ∗ - B - B ∗ | | Ψ ⩽ | | L [ μ ] Δ A ∗ , A (I) - L [ μ ] Δ C ∗ , C (I) (A X + X B - C X - X D) × L [ μ ] Δ B , B ∗ (I) - L [ μ ] Δ D , D ∗ (I) | | Ψ , holds under any of the following conditions: (a) if (b) if for some p ⩾ 2 , L 2 ( R + , μ) is separable and at least one of pairs (A, C) or (B, D) consists of normal operators, (c) if both pairs (A, C) and (B, D) consist of normal operators. Above, Φ ( p ) denotes (the degree) p-modified s.n. function Φ and Φ ( p ) ∗ is the dual s.n. function for Φ ( p ) . Moreover, the aforementioned inequality is generalized to the iterated perturbations of Laplace transformers, and the alternative inequalities are given for Q norms as well. These inequalities also generalize some previously obtained results. [ABSTRACT FROM AUTHOR]

We introduce and study some new uniform structures for Hilbert C ∗ -modules over a C ∗ -algebra A. In particular, we prove that in some cases they have the same totally bounded sets. To define one of them, we introduce a new class of A -functionals: locally adjointable functionals, which have interesting properties in this context and seem to be of independent interest. A relation between these uniform structures and the theory of A -compact operators is established. [ABSTRACT FROM AUTHOR]

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain Ω ⊂ R d . This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of ℓ r type, 1 ≤ r ≤ ∞ . [ABSTRACT FROM AUTHOR]

For 1 ≤ p < ∞ , let L p (M , τ) be the non-commutative L p -space associated with a von Neumann algebra M , where M admits a normal semifinite faithful trace τ . Using the trace τ , Banach duality formula and Gâteaux derivative, this paper characterizes an element a ∈ L p (M , τ) such that ‖ a ‖ p = inf { ‖ a + b ‖ p : b ∈ B p } , where B p is a closed linear subspace of L p (M , τ) and ‖ · ‖ p is the norm on L p (M , τ) . Such an a is called B p -minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces B p = ⨁ i = 1 ∞ e i S e i (converging with respect to ‖ · ‖ p ) are considered, where { e i } i = 1 ∞ is a sequence of mutually orthogonal and τ -finite projections in a σ -finite von Neumann algebra M , and S is the set of elements in M with τ -finite supports. [ABSTRACT FROM AUTHOR]

Let A be a bounded linear operator in a complex separable Hilbert space ℌ, and S be a selfadjoint operator in ℌ. Assuming that A − S belongs to the Schattenvon Neumann ideal S p (p > 1) , we derive a bound for ∑ k ∣ R λ k (A) − λ k (S) ∣ p , where λk(A) (k = 1, 2, ...) are the eigenvalues of A. Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. In addition, in the case p = 2 we refine the Weyl inequality between the real parts of the eigenvalues of A and the eigenvalues of its Hermitian component. [ABSTRACT FROM AUTHOR]

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles. [ABSTRACT FROM AUTHOR]

The main aim of this work is to show the solvability of the homogeneous and inhomogeneous boundary conjugation problems for piecewise analytic functions in the Besov space, which is embedded into the class of continuous functions. [ABSTRACT FROM AUTHOR]

In this article, we derive the restriction theorem for the special Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions associated with the special Hermite operator. Further, we discuss the optimal behavior of the constant as a limit of a large number of functions. [ABSTRACT FROM AUTHOR]

Let M be a von Neumann algebra with a normal faithful semifinite trace. In this paper, we consider that in n-tuples of noncommutative L p -spaces l s (n) (L p (M)) , the norm is invariant under the action of invertible elements in M . Then we prove that the complex interpolating theorem in the case of l s (n) (L p (M)) . Using this result, we obtain that Clarkson’s inequalities for n-tuples of operators with weighted norm of noncommutative L p -spaces, where the weight being a positive invertible operator in M . [ABSTRACT FROM AUTHOR]

We prove that a normalised linear functional on certain symmetric spaces associated with a semifinite von Neumann algebra, respects tail majorisation if and only if it is a Dixmier-type trace. [ABSTRACT FROM AUTHOR]

Influenced by the concept of a p-compact operator due to Sinha and Karn (Stud Math 150(1): 17–33, 2002), we introduce p-compact Bloch maps of the open unit disk D ⊆ C to a complex Banach space X, and obtain its most outstanding properties: surjective Banach ideal property, Möbius invariance, linearisation on the Bloch-free Banach space over D , inclusion properties, factorisation of their derivatives, and transposition on the normalized Bloch space. We also present right p-nuclear Bloch maps of D to X and study its relation with p-compact Bloch maps. [ABSTRACT FROM AUTHOR]

Let ∑ i = 1 ∞ A i A i ∗ and ∑ i = 1 ∞ A i ∗ A i converge in the strong operator topology. We study the map Φ A defined on the Banach space of all bounded linear operators B (H) by Φ A (X) = ∑ i = 1 ∞ A i X A i ∗ and its restriction Φ A | K (H) to the Banach space of all compact operators K (H). We first consider the relationship between the boundary eigenvalues of Φ A | K (H) and its fixed points. Also, we show that the spectra of Φ A and Φ A | K (H) are the same sets. In particular, the spectra of two completely positive maps involving the unilateral shift are described. [ABSTRACT FROM AUTHOR]

Let S p be the p-Schatten–von Neumann ideal of operators in a Hilbert space, A ∈ S p (2 ≤ p < ∞) , A R = (A + A ∗) / 2 , A I = (A - A ∗) / (2 i) and λ k (A) (k = 1 , 2 ,...) are the eigenvalues of A. The asterisk means the adjointness. We derive the inequality ∑ k = 1 ∞ | I m λ k (A) - λ k (A I) | p ≤ η p N p p (A R) , where η p ≤ 2 p is an explicitly defined constant dependent on p, only, and N p (.) is the norm of S p . In addition, assuming that A I is a nuclear operator, we obtain an inequality for the series of the imaginary parts of the eigenvalues of A via the trace of A I . [ABSTRACT FROM AUTHOR]

In this short note, the second dual of generalized group algebra (ℓ 1 (G , A) , ∗) equipped with both Arens product is investigated, where G is any discrete group and A is a Banach algebra containing a complemented algebraic copy of (ℓ 1 (N) , ∙) . We give an explicit family of annihilators(w.r.t both the Arens product) in the algebra ℓ 1 (G , A) ∗ ∗ , arising from non-principal ultrafilters on N and which are not in the toplogical center. As a consequence, we also deduce the fact that ℓ 1 (G , A) is not Strongly Arens irregular. [ABSTRACT FROM AUTHOR]

A Hilbert space operator $ A\in \mathcal {B(H)} $ A ∈ B (H) is a generalized n-projection, $ A\in (G-n-P) $ A ∈ (G − n − P) , if $ {A^*}^n=A $ A ∗ n = A. The product AB of a commuting pair A, B of $ (G-n-P) $ (G − n − P) -operators is a $ (G-n-P) $ (G − n − P) -operator. The converse fails. We prove that if $ \|AB\|=\|A\|\|B\| $ ‖ AB ‖ = ‖ A ‖ ‖ B ‖ and $ \sigma _a(AB)=\sigma _a(A)\sigma _a(B) $ σ a (AB) = σ a (A) σ a (B) , then $ AB\in (G-n-P) $ AB ∈ (G − n − P) implies $ {\tfrac {A}{\|A\|}}, {\tfrac {B}{\|B\|}} $ A ‖ A ‖ , B ‖ B ‖ are $ (G-n-P) $ (G − n − P) if and only if A and B are normal operators. Translated to tensor products (and upon identifying the tensor product $ A\otimes B $ A ⊗ B with the left-right multiplication operator $ {\mathcal {E}}_{A,B^*} $ E A , B ∗ acting on the Hilbert–Schmidt bimodule $ {\mathcal {C}}_2(\mathcal {H}) $ C 2 (H)) this says that $ A\otimes B $ A ⊗ B (resp., $ {\mathcal {E}}_{A,B^*} $ E A , B ∗ ) is a $ (G-n-P) $ (G − n − P) -operator implies $ {\tfrac {A}{\|A\|}}, {\tfrac {B}{\|B\|}} $ A ‖ A ‖ , B ‖ B ‖ are $ (G-n-P) $ (G − n − P) if and only if A and B are normal operators. [ABSTRACT FROM AUTHOR]

In this article, we address the following Strichartz’s restriction problem: For a given surface S embedded in R n × R d with n + d ≥ 2 , for what values of 1 ≤ p < 2 , do we have ∫ S | f ^ (ξ , ζ) | 2 h κ 2 (ζ) d σ (ξ , ζ) 1 2 ≤ C ‖ f ‖ L κ p (R n × R d) ? Here f ^ is the Fourier–Dunkl transform of f (defined in (1.2)). In particular, we prove Strichartz’s restriction theorem for the Fourier–Dunkl transform for certain surfaces, namely, paraboloid, sphere, and hyperboloid, and its generalization to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish respected versions of Strichartz estimates for Schrödinger’s propagator and Klein–Gordon propagator associated with the Dunkl Laplacian. This restriction theorem generalizes the Stein-Tomas and Strichartz’s restrictions theorems in the special cases. [ABSTRACT FROM AUTHOR]

We characterize the integration operators V g with symbol g for which V g acts as an absolutely summing operator on weighted Bloch spaces B β and on weighted Bergman spaces A α p . We show that V g is r-summing on A α p , 1 ≤ p < ∞ , if and only if g belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators V g on Bloch spaces and on Bergman spaces. [ABSTRACT FROM AUTHOR]

In this work, we give several new upper and lower bounds for the p-numerical radii of operators as well as for n × n operator matrices. An application to 2-nilpotent operators is provided, and a p-numerical radius power inequality is also given. [ABSTRACT FROM AUTHOR]

The geometry of the compact convex set of all n×nn\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓnp{\ell }_{n}^{p} to ℓnp{\ell }_{n}^{p} and the Schatten pp-norms, both for the range 1≤p≤∞1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.

The aim of the article is to show the invalidity of the Strichartz estimate for the free Schrödinger equation associated with the Ornstein–Uhlenbeck operator L = - 1 2 Δ + ⟨ x , ∇ ⟩ in R n . As a consequence we obtain a negative answer to the Fourier–Hermite restriction problem associated with the normalized Hermite polynomials for a discrete surface in Z × N 0 n. [ABSTRACT FROM AUTHOR]

In this work, we give some new p-numerical radius inequalities for 2 × 2 operator matrices. [ABSTRACT FROM AUTHOR]

We characterize the Hilbert–Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class Sp result for dyadic paraproducts, where 1 < p < ∞ . We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform. [ABSTRACT FROM AUTHOR]

In this work, we study a comparison of norms in non-commutative spaces of τ -measurable operators associated with a semifinite von Neumann algebra. In particular, we obtain Nazarov–Podkorytov type lemma Nazarov et al. (Complex analysis, operators, and related topics. Operatory theory: advances, vol 113, pp 247–267, 2000) and extend the main results in Astashkin et al. (Math Ann, 2023. ) to non-commutative settings. Moreover, we complete the range of the parameter p for 0 < p < 1. [ABSTRACT FROM AUTHOR]

Let Ψ , Φ be s.n. functions, p ≥ 2 , and let φ be an operator monotone function on [ 0 , ∞) such that φ (0) = 0. If are such that A and B are strictly accretive and then also and | | A X φ (B) - φ (A) X B | | Ψ ≤ φ ( A + A ∗ 2 ) - A + A ∗ 2 φ ′ ( A + A ∗ 2 ) ( A + A ∗ 2 ) - 1 A (A X - X B) B ( B + B ∗ 2 ) - 1 φ ( B + B ∗ 2 ) - B + B ∗ 2 φ ′ ( B + B ∗ 2 ) Ψ . under any of the following conditions: Both A and B are normal, A is cohyponormal, B is hyponormal and at least one of them is normal, and Ψ : = Φ (p) ∗ , A is cohyponormal, B is hyponormal and | | · | | Ψ is the trace norm | | · | | 1. Alternative inequalities for | | · | | Φ (p) norms are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we define and study a vector-valued sequence space, called the space of anisotropic (s, q, r)-summable sequences, that generalizes the classical space of (s; q)-mixed sequences (or mixed (s; q)-summable sequences). Furthermore, we define two classes of linear operators involving this new space and one of them generalizes the class of (s; q) -mixed linear operators due A. Pietsch. Some characterizations, inclusion results and a Pietsch domination-type theorem are presented for these classes. It is worth mentioning that some of these results are new even in the particular cases of mixed summable sequences and mixed summing operators. [ABSTRACT FROM AUTHOR]

We characterize the L p – L q compactness of Bergman type operators, which are singular integral operators induced by the modified Bergman kernel on the complex unit ball. Moreover, we characterize Schatten class and Macaev class Bergman type integral operators on the Lebesgue space and the Bergman space by the methods of spectral estimates and operator inequalities; we also give a relatively intrinsic characterization by introducing a concept of dimension of a compact operator. The Dixmier trace of Bergman type operators is also calculated. [ABSTRACT FROM AUTHOR]

We introduce a new norm on C 2 × C 2 , where C 2 is the Hilbert-Schmidt class. We study basic properties of this norm and prove inequalities involving it. As an application of the present study, we deduce a chain of new bounds for the Hilbert-Schmidt numerical radii of 2 × 2 operator matrices. Connection with the classical Hilbert-Schmidt numerical radius of a single operator are also provided. Moreover, we refine some related existing bounds, too. [ABSTRACT FROM AUTHOR]

(1) Suppose μ is a smooth measure on a smooth hypersurface of positive Gaussian curvature in R 2 n . If n ≥ 2 , then W (μ) , the Weyl transform of μ is a compact operator, and if p > n ≥ 6 , then W (μ) belongs to the p-Schatten class. (2) There exist Schatten class operators with linearly dependent quantum translates. [ABSTRACT FROM AUTHOR]

We analyse the problem of characterizing optimal dual frames for erasures with respect to the spectral radius. Sufficient conditions are obtained for a dual frame of a given frame to be an optimal dual frame for two erasures. Sufficient conditions are also obtained for the canonical dual to be a two-erasure optimal dual with respect to spectral radius. We also establish some relations for optimality conditions between one erasure and two erasures in spectrally optimal dual frames. [ABSTRACT FROM AUTHOR]

In this paper, we establish the Schatten class and endpoint weak Schatten class estimates for the commutator of Riesz transforms on weighted L2 spaces. As an application a weighted version for the estimate of the quantised derivative introduced by Alain Connes and studied recently by Lord–McDonald–Sukochev–Zanin and Frank–Sukochev–Zanin is provided. [ABSTRACT FROM AUTHOR]

We study nuclear embeddings for function spaces of generalised smoothness defined on a bounded Lipschitz domain Ω ⊂ ℝd. This covers, in particular, the well-known situation for spaces of Besov and Triebel–Lizorkin spaces defined on bounded domains as well as some first results for function spaces of logarithmic smoothness. In addition, we provide some new, more general approach to compact embeddings for such function spaces, which also unifies earlier results in different settings, including also the study of their entropy numbers. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) about nuclear diagonal operators acting in ∓r spaces, which we could recently extend to the vector-valued setting needed here. [ABSTRACT FROM AUTHOR]

Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1–34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on L 2 (R d) . We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen’s class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned. [ABSTRACT FROM AUTHOR]