In this article, we develop a framework for the joint functional calculus of commuting pair of Ritt_E operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for pair of Ritt_E operators to that of their associated sectorial counterparts. In addition, we prove a joint dilation theorem for commuting pair of Ritt_E operators on a broad class of Banach spaces. As a key application, we obtain an equivalent set of criteria on L^p -spaces for 1< p < ∞ that determine when a commuting pair of Ritt_E operators admits a joint bounded functional calculus.

In this paper, we develop a novel framework for quantitative mean ergodic theorems in the noncommutative setting, with a focus on actions of amenable groups and semigroups. We prove square function inequalities for ergodic averages arising from actions of groups of polynomial volume growth on a fixed noncommutative $L_p$-space for $1<p<\8$. To achieve this, we establish two endpoint estimates for a noncommutative square function on non-homogeneous space. Our approach relies on semi-commutative non-homogeneous harmonic analysis, including the non-doubling Calderón-Zygmund arguments for non-smooth kernels and $\mathrm{BMO}$ space theory, operator-valued inequalities related to balls and cubes in groups equipped with non-doubling measures, and a noncommutative generalization of the classical transference method for amenable group actions. As an application, we establish a quantitative ergodic theorem for the ergodic averages associated with the positive power of modulus representation arising from a Lamperti representation on noncommutative $L_p$-spaces, extending some results in \cite{Templeman2015}. To obtain quantitative ergodic theorem for semigroups of operators, in this paper, we address the open question of extending dilation theorem of Fackler-Glück from single operators to commuting tuples on Banach spaces including noncommutative $L_p$-spaces. Indeed our approach provides genuine joint $N$-dilations for commuting families, unifying and extending the classical dilation theorems of Sz.-Nagy--Foiaş and Akçoglu--Sucheston for a natural class of commuting tuple of contractions extending the abstract dilation theorem of of Fackler-Glück for commuting tuple of contractions. This enables us to obtain a quantitative ergodic theorem for a large class of semigroups of operators on $\mathbb{R}^d_{+}$.

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. Let $1<p<\infty.$ In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all $x \in L_p(\mathcal{M})$, extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, the similar inequalities as above hold for arbitrary $L_\infty-L_1$ contractions with the same condition on the numerical range, yielding a vast generalisation of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. We further prove a variational inequality for contractively regular operators $T:L_p(Ω)\to L_p(Ω)$ whose peripheral spectrum is finite and satisfies a suitable resolvent estimate, extending earlier work of Le Merdy and Xu \cite{le-Merdy-Xu-q-variational-inequality}. Finally, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Below \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative $(p,p)$-maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalisation of Stein's method of embedding maximal function into analytic family of operators.

We study several interrelated problems arising from the interplay between extreme point theory, Grothendieck-type inequalities, and tensor product norms. We develop a general framework for characterizing the extreme points of the set of positive contractions 𝒜_X→ Y between finite-dimensional Banach spaces, with explicit results for X=ℓ_1^n, Y=ℓ_∞^n and vice versa. These characterizations are applied to evaluate several constants exactly. We show that the positive Grothendieck constant K_G^+,ℝ(3) equals 9/8 and that the smallest constant ρ^+(X) for which A_π⩽ ρ^+(X)A_ε holds for all A ⩾ 0 equals 5/4 when X=ℓ^3_∞(ℝ). We also prove that ρ^+(X)=1 when X=ℓ_∞^n(ℂ) and n⩽ 3. Finally, we prove that ρ^+(X) = 1 for every 2-dimensional subspace X of ℓ^3_∞(ℂ); since this is stronger than the 2-summing property, it recovers Proposition 4.4 of .

In this paper, we obtain the desired noncommutative maximal inequalities of the truncated Calderón–Zygmund operators of non-convolution type acting on operator-valued Lp-functions for all 1 < p < ∞, answering a question left open in the previous work [11].

In this article, we develop a framework for the joint functional calculus of commuting tuples of $\text{Ritt}_{\text{E}}$ operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for tuples of $\text{Ritt}_{\text{E}}$ operators to that of their associated sectorial counterparts. In addition, we prove a joint dilation theorem for commuting tuples of $\text{Ritt}_{\text{E}}$ operators on a broad class of Banach spaces. As a key application, we obtain an equivalent set of criteria on $L^p$-spaces for $1

Let G be a connected simple Lie group of real rank one and finite center, and let K be a maximal compact subgroup. We study the families of spherical, ball, and uniform averages (σ_t)_t>0, (β_t)_t>0, and (μ_t)_t>0 on G induced by the canonical G-invariant metric on G/K, in the setting where G acts by trace-preserving *-automorphisms on a finite von Neumann algebra (ℳ,τ). For the associated noncommutative L_p-spaces L_p(ℳ), we consider both local and global noncommutative maximal inequalities for these averages, and corresponding pointwise ergodic theorems in the sense of bilateral almost uniform convergence. Our approach combines a noncommutative Calderón transfer principle, spectral analysis for the Gelfand pair (G,K) via Harish–Chandra's spherical functions, fractional integration methods, and Littlewood–Paley g-function estimates. This work is complemented by our results for higher-rank semisimple Lie groups, where the presence of Property (T) and associated spectral gaps serve as the key tools in establishing Wiener-type noncommutative pointwise ergodic theorems and noncommutative L_p-maximal inequalities for ball and spherical averages on G/K for certain class of semisimple Lie groups.

The existence of isometric embedding of $S_q^m$ into $S_p^n$ , where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$ , has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$ . More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$ , where $(q,p)\in (0,\infty )\times (0,1)$ and $p\neq q$ . As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$ , where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ $\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$ . Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$ , where $(q,p)\in [2, \infty )\times (0,1)$ . A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative Lp$L_p$ ‐spaces for a fixed 1

AbstractIn this article, we study the following question asked by Michael Hartz in a recent paper (Every complete Pick space satisfies the column-row property, to appear in Acta Mathematica): which operator spaces satisfy the column–row property? We provide a complete classification of the column–row property (CRP) for noncommutative $L_{p}$ -spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the CRP and discuss their existence and nonexistence for various natural examples of operator spaces.

We investigate a Grothendieck-type inequality for pairs of Banach spaces E,F assuming E is finite-dimensional and study the associated Grothendieck-type constant. We prove that if there is a C >0 such that A⊗id_F_E_m⊗̌F→ E_n^*⊗̂F⩽ C A_E_m→ E_n^* for all m,n∈ℕ, where E_n=n, then both F and F^* must have finite cotype. Moreover, assuming that F has the bounded approximation property and that the conjecture in has an affirmative answer, we show that (E_n^*)_n⩾ 1 satisfies G.T. uniformly. We show that the Grothendieck-type constant defined for a pair of Banach spaces (E,F) is closely related to another interesting quantity introduced recently in comparing the projective and injective norms on the tensor product of two finite-dimensional Banach spaces E and F. We also study analogously the constants appearing in these extremal problems by restricting only to non-negative tensors. For contractive little Parrott homomorphisms ϱ_V : H^∞(Ω) → M_n, where Ω is the dual unit ball of a finite dimensional Banach space (E,·), we prove the sharp estimate ϱ_V_cb≤√(γ(E)), γ(E) being the positive Grothendieck constant associated with the pair (E, ℓ^n_2).

In this paper we propose a generalization of the Grothendieck inequality for pairs of Banach spaces $E$ and $F$ with $E$ being finite dimensional and investigate the behaviour of the Grothendieck constant $K_G(E,F)$ implicit in such an inequality. We show that if $\sup\{K_G(E_n,F): n\geqslant 1\}$ is finite for some sequence of finite dimensional Banach spaces $(E_n)_{n\geqslant 1}$ with $\dim E_n=n$, and an infinite dimensional Banach space $F$, then both $F$ and $F^*$ must have finite cotype. In addition to that if $F$ has the bounded approximation property, we conclude that $(E_n^*)_{n\geqslant 1}$ satisfies G.T. uniformly by assuming the validity of a conjecture due to Pisier. We also show that $K_G(E,F)$ is closely related to the constant $\rho(E,F)$, introduced recently, comparing the projective and injective norms on the tensor product of two finite dimensional Banach spaces $E$ and $F$. We also study, analogously, these constants by computing the supremum only on non-negative tensors.

In this paper, we study existence of isometric embedding of Sqm into Spn, where 1≤p≠q≤∞ and n≥m≥2. We show that for all n≥m≥2 if there exists a linear isometry from Sqm into Spn, where (q,p)∈(1,∞]×(1,∞)∪(1,∞)∖{3}×{1,∞} and p≠q, then we must have q=2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever Sq embeds isometrically into Sp for (q,p)∈(1,∞)×[2,∞)∪[4,∞)×{1}∪{∞}×(1,∞)∪[2,∞)×{∞} with p≠q, we must have q=2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative Lp-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m≥2 and 1<q<2, Sqm embeds isometrically into S∞n, was left open in Bull. London Math. Soc. 52 (2020) 437-447.

. The existence of isometric embedding of S mq into S np , where 1 ≤ p 6 = q ≤ ∞ and m, n ≥ 2 has been recently studied in [6]. In this article, we extend this study beyond the above mentioned range of p and q , and hence in the context of quasi-Banach space setting. A new tool in our paper is the non-commutative Clarkson’s inequality for Schatten class operators. Other tools involved are the Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

. The existence of isometric embedding of S mq into S np , where 1 ≤ p 6 = q ≤ ∞ and m, n ≥ 2 has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above mentioned range of p and q . More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space ℓ mq ( R ) into ℓ np ( R ), where ( q, p ) ∈ (0 , ∞ ) × (0 , 1) and p 6 = q . As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of S mq into S np , where ( q, p ) ∈ (0 , 2) × (0 , 1) ∪ { 1 } × (0 , 1) \ { 1 n : n ∈ N } ∪ {∞} × (0 , 1) \ { 1 n : n ∈ N } and p 6 = q . Moreover, in some restrictive cases, we also show that there is no isometric embedding of S mq into S n p , where ( q, p ) ∈ [2 , ∞ ) × (0 , 1). To achieve our goal we significantly use Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

We study existence of linear isometric embedding of ℓpm into S∞ , for 1⩽p<∞ , and unique operator space structure on two‐dimensional Banach spaces. For p∈(2,∞)∪{1} , we show that indeed ℓp2 does not embed isometrically into S∞ . This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza (Houston J. Math. 44 (2018) 1205–1212). We also show that S1m does not embed isometrically into Spn for all 1

In this article, we study multivariate generalizations of Matsaev’s conjecture in commutative and non-commutative $$L^p$$-spaces. We prove that the multivariate analogue of Matsaev’s conjecture is eventually false for all $$1<p<\infty .$$ We exhibit various joint dilation results on non-commutative $$L^p$$-spaces.

In this paper, we study existence of isometric embedding of $S_q^m$ into $S_p^n$ where $1\leq p\neq q\leq \infty$ and $m,n\geq 2.$ We show that if there is a linear isometry from $S_q^m$ into $S_p^n$, where $(q,p)\in(1,\infty]\times(1,\infty)\cup(1,2)\times\{\infty\}$ and $p\neq q$ then we must have $q=2.$ This mostly generalizes a classical result of Lyubich and Vasserstein. Moreover, our work complements work of Junge, Xu, Parcet and others on isometric and almost isometric embedding theory on noncommutative $L_p$-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The case $1

We study existence of linear isometric embedding of ℓ p m into S ∞ , for 1 ⩽ p < ∞ , and unique operator space structure on two-dimensional Banach spaces. For p ∈ ( 2 , ∞ ) ∪ { 1 } , we show that indeed ℓ p 2 does not embed isometrically into S ∞ . This verifies a guess of Pisier and broadly generalizes the main result of Gupta and Reza ( Houston J. Math . 44 (2018) 1205–1212). We also show that S 1 m does not embed isometrically into S p n for all 1 < p < ∞ and m ⩾ 2 . As a consequence, we establish non-commutative analogue of some of the results of Lyubich and Shatalova ( Algebra i Analiz 16 (2004) 15–32). We also show that ( C 2 , ∥ . ∥ B p , q ) does not embed isometrically into S ∞ for 2 < p , q < ∞ . The main ingredients in our proofs are notions of Birkhoff–James orthogonality and norm-parallelism for operators on Hilbert spaces. These enable us to deploy ‘infinite descent’ type of arguments to obtain contradictions. Our approach is new even in the commutative case. We prove that ( C 2 , ∥ . ∥ B p , q ) does not have unique operator space structure whenever ( p , q ) ∈ ( 1 , ∞ ) × [ 1 , ∞ ) ∪ [ 1 , ∞ ) × ( 1 , ∞ ) by showing that they do not have Property P or two summing property. In view of Misra, Pal and Varughese ( J. Operator Theory 82 (2019) 23–47), this produces genuinely new examples of two-dimensional Banach spaces without unique operator space structure, providing a partial answer to a question of Paulsen. In this case, we derive our result by transferring the problem to real case and applying known results of Arias, Figiel, Johnson and Schechtman ( Trans. Amer. Math. Soc . 347 (1995) 3835–3857).