# Reactive and human-in-the-loop planning and control of multi-robot systems
under LTL specifications in dynamic environments*

Pian Yu1, Gianmarco Fedeli2, and Dimos V. Dimarogonas3 *This work was
partially supported by the Swedish Research Council (VR), the Knut and Alice
Wallenberg Foundation (KAW), the EIC Horizon Europe SymAware, the EU project
CANOPIES, the ERC COG LEAFHOUND (Grant agreement ID: 864720), and the ERC ADG
FUN2MODEL (Grant agreement ID: 834115). 1Pian Yu is with the Department of
Computer Science, Oxford University, Oxford, United Kingdom
<EMAIL_ADDRESS>Gianmarco is with Bosch, Braga, Portugal
<EMAIL_ADDRESS>Dimarogonas is with the Division of Decison
and Control Systems, KTH, Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract

This paper investigates the planning and control problems for multi-robot
systems under linear temporal logic (LTL) specifications. In contrast to most
of existing literature, which presumes a static and known environment, our
study focuses on dynamic environments that can have unknown moving obstacles
like humans walking through. Depending on whether local communication is
allowed between robots, we consider two different online re-planning
approaches. When local communication is allowed, we propose a local trajectory
generation algorithm for each robot to resolve conflicts that are detected on-
line. In the other case, i.e., no communication is allowed, we develop a model
predictive controller to reactively avoid potential collisions. In both cases,
task satisfaction is guaranteed whenever it is feasible. In addition, we
consider the human-in-the-loop scenario where humans may additionally take
control of one or multiple robots. We design a mixed initiative controller for
each robot to prevent unsafe human behaviors while guarantee the LTL
satisfaction. Using our previous developed ROS software package, several
experiments are conducted to demonstrate the effectiveness and the
applicability of the proposed strategies.

##  I INTRODUCTION

During the past decade, there is a surge of using temporal logic formulas,
such as linear temporal logic (LTL) [1], to concisely specify desired
behaviors of robotic systems [2, 3, 4, 5, 6, 7, 8, 9, 10]. This is due to the
expressiveness of LTL formulas in capturing many common robotic tasks, e.g.,
ordered reachability, collision avoidance, surveillance, and ordered supply
delivery [11]. In addition, the availability of automated toolboxes [1, 12]
make the use of LTL more appealing. The planning and control under LTL
specifications have been extensively investigated for a single robot [2, 3, 4,
5] and multi-robot systems (MRSs) [6, 7, 8, 9, 10]. However, most of the
existing literature assume that the environment is static and known a priori.
When the environment is dynamic (e.g., there are moving obstacles like humans
walking through), for which online replanning becomes a necessity, the problem
of safe and efficient replanning under LTL specifications becomes challenging
during the online execution of robots.

Along the rapid advancement of automation technology, the past decade has
witnessed a growing emphasis on human-robot collaboration. Indeed, many
autonomous systems are performing their designated tasks while being
supervised or collaborating with human operators [13]. On one hand, having a
human-in-the-loop is useful for guiding the robot through challenging tasks.
On the other hand, the possibly erroneous inputs from the human can be
dangerous for the autonomous systems. Therefore, effectively managing real-
time interactions between the autonomous system and humans is crucial for
ensuring the safety and efficiency of the entire system.

When a single robot is considered, authors in [14] propose a plan revising
mechanism assuming the environment is static and partially known. Once a
transition in the current plan becomes invalid, it finds the shortest path
bridging up the two components. The work [15] develops an iterative repair
strategy to resolve unknown obstacles by combining local patching with refined
triangulation. However, this strategy cannot deal with moving obstacles. For
the case of MRSs, most of research assumes that the MRS adheres to a global
LTL specification, subsequently addressing an offline motion planning problem
through a centralized approach [6, 7]. In [16], MRSs under local LTL
specifications are studied and a distributed motion coordination algorithm is
proposed to resolve conflicts. In this work, the environment is static and
local communication between robots is required for conflict resolution.
Furthermore, when humans are involved in the robot control, the concept of a
mixed-initiative controller (MIC) is introduced in [17], which integrates
external human inputs with the conventional navigation controller. This method
is further investigated in [18], where the human initiative influences both
the high level tasks and the low level continuous inputs.

This work aims to develop reactive planning and control algorithms for MRSs
under LTL specifications. First, we build upon our prior research [16] by
addressing dynamic environments, which present a non-trivial challenge.
Additionally, when humans participate in controlling the robots, we expand the
MIC proposed in our previous work [18] to accommodate MRSs. The contributions
are threefold: i) Depending on whether local communication is allowed between
robots, we propose two different online re-planning approaches for MRSs
operating in dynamic environments. In the first scenario, where local
communication among robots is allowed, we propose a local trajectory
generation algorithm to resolve conflicts. In the second scenario, where there
is no communication between the robots, we develop a model predictive
controller to reactively handle potential collisions. In both cases task
satisfaction is guaranteed whenever it is feasible. ii) When one or multiple
robots are controlled by human operators, a MIC is designed for the MRSs to
guarantee safety and LTL task satisfaction. iii) Several experiments are
carried out to demonstrate the effectiveness of the proposed strategies.

##  II Preliminaries and Problem Formulation

###  II-A Linear temporal logic (LTL)

We use LTL to concisely specify the desired robot behaviour. LTL is built from
a set of atomic propositions $AP$, the logic connectives of negation ($\neg$),
conjunction ($\wedge$), and disjunction ($\wedge$), and the temporal operators
next ($\bigcirc$), until ($\mathsf{U}$), eventually ($\lozenge$), and always
($\square$). An LTL formula is formed inductively according to the following
syntax [1]:

$\varphi::=\top|a|\neg\varphi|\varphi_{1}\wedge\varphi_{2}|\bigcirc\varphi|\varphi_{1}\mathsf{U}\varphi_{2},$
(1)

where $a\in AP,\varphi,\varphi_{1},\varphi_{2}$ are LTL formulas. The logic
connective $\vee$ and temporal operators $\lozenge$ and $\square$ can be
derived inductively. We omit the full LTL semantics due to space limitations
and refer the reader to [1] for details. The satisfaction of an LTL formula
$\varphi$ over $AP$ can be captured through a nondeterministic Büchi automaton
(NBA) [19], defined as a tuple $\mathsf{B}=(S,S_{0},2^{AP},\delta,F)$, where

  * •

$S$ is a finite set of states,

  * •

$S_{0}\subseteq S$ is the set of initial states,

  * •

$2^{AP}$ is the input alphabet,

  * •

$\delta:S\times 2^{AP}\to 2^{S}$ is the transition function, and

  * •

$F\subseteq S$ is the set of accepting states.

An infinite _run_ $\bm{s}=s_{0}s_{1}\ldots$ is called _accepting_ if
$\texttt{Inf}(s)\cap F\neq\emptyset$, where $\texttt{Inf}(s)$ is the set of
states that appear in $\bm{s}$ infinitely often. Several translation tools,
e.g., LTL2BA [20], are available to obtain $\mathsf{B}$ given $\varphi$.

###  II-B Robot dynamics and motion abstraction

Consider a set of $N$ robots navigating within a bounded workspace
$\mathcal{W}$. The dynamics of robot $i$ is given by:

$\displaystyle\dot{x}_{i}(t)$ $\displaystyle=v_{i}(t)\cos(\theta_{i}(t)),$ (2)
$\displaystyle\dot{y}_{i}(t)$ $\displaystyle=v_{i}(t)\sin(\theta_{i}(t)),$
$\displaystyle\dot{\theta}_{i}(t)$ $\displaystyle=w_{i}(t),$

where $p_{i}=[x_{i},y_{i}]^{T}$ is the Cartesian position, $\theta_{i}$ is the
orientation, and $u_{i}=[v_{i},w_{i}]^{T}$ is the input vector of robot $i$.
The input of robot $i$ is constrained to the compact set $\mathcal{U}_{i}$,
i.e., $u_{i}(t)\in\mathcal{U}_{i},\forall t\geq 0.$

Within the workspace $\mathcal{W}$, there is a set of properties (atomic
propositions) $AP=\left\\{a_{1},a_{2},\cdots,a_{M}\right\\}$, e.g. ”this is a
goal region”, “this is a charging station”. Let
$\Phi:=\\{X_{1},\ldots,X_{M}\\}$ be a partition of $\mathcal{W}$ such that
$\mathcal{W}=\cup_{l=1}^{M}X_{l}$. Define $L:\mathcal{W}\to 2^{AP}$ as a
labelling function. Given a point $p\in\mathcal{W}$, define the function
$Q:\mathcal{W}\to\Phi$ as $Q(p):=\\{X_{l}\in\Phi:p\in X_{l}\\}.$ It maps a
state $p$ into the region $X_{l}$ that contains it. We say $\Phi$ is an
_observation preserving_ partition of $\mathcal{W}$ if it satisfies

$L(p)=L(p^{\prime}),\forall p,p^{\prime}:Q(p)=Q(p^{\prime}).$

Given an observation preserving partition of the workspace, the dynamics of
robot $i$ (2) can be abstracted as a controlled transition system (CTS)
$\mathcal{T}_{i}=(\Phi,\phi_{0,i},\mathcal{U}_{i},AP,\rightarrow_{i},L)$,
where $\phi_{0,i}\in\Phi$ is the initial state and
$\rightarrow_{i}:\Phi\times\mathcal{U}_{i}\to\Phi$ is the transition relation.

###  II-C Objectives

We assign an LTL specification $\varphi_{i}$ to each robot $i$, which is
specified over the set of atomic propositions $AP$. Nevertheless, the
workspace is dynamic, for instance, there may be moving obstacles like humans
walking through. In addition, it is assumed that each robot has only limited
sensing capabilities, that is, the robot can detect other robots or the
unknown moving obstacles only if they are within its sensing region
$\mathcal{B}(p_{i},R_{i})$, where $p_{i}$ is the position and $R_{i}$ is the
sensing radius of robot $i$. The initial trajectory planning (which is
detailed in the next section) cannot take into account these unpredictable
situations, and thus online replanning is necessary for each robot to
guarantee safe operation and task satisfaction.

Given the abstract model, i.e., the CTS $\mathcal{T}_{i}$, and the task
specification $\varphi_{i}$ of each robot $i$, the first objective (O1) is to
design a planning algorithm for each robot $i$ such that the LTL task
$\varphi_{i}$ is satisfied and safety is always guaranteed despite of the
dynamic environments. In addition, we further consider the human-in-the-loop
context, where human operators can take over the control of the robots from
the on-board autonomous controller. The second objective (O2) is to design
control algorithm for each robot $i$, which can react to (possibly dangerous)
human inputs while preserving safety and task satisfaction.

##  III Online Replanning in dynamic environments

Before implementation, an initial satisfying plan needs to be generated for
each robot $i$. This procedure is implemented in the package LTL core &
planner [21]. It is based on constructing a product Büchi automaton (PBA)
$\mathcal{P}_{i}$ between the CTS $\mathcal{T}_{i}$ and the NBA
$\mathsf{B}_{i}$ (which is translated from $\varphi_{i}$). Using model-
checking methods [12], an accepting run can be obtained from the PBA
$\mathcal{P}_{i}$ and projected back to the CTS $\mathcal{T}_{i}$ intersection
$\mathsf{B}_{i}$. Accepting runs have a _prefix-suffix_ structure of this
kind: $r_{\mathcal{P}_{i}}=p_{0},p_{1}\cdots p_{k}\left(p_{k+1}\ldots\ldots
p_{n}p_{k}\right)^{\omega}$. The output word is composed of two separate
parts: a finite prefix that is executed only once from the initial state
$p_{0}$ to an accepting state $p_{k}$ and a suffix that is repeated infinitely
from the accepting state $p_{k}$ to itself. The accepting run has a
corresponding action sequence that each robot must carry out in a prefix-
suffix structure in order to fulfill $\varphi_{i}$.

Note that the initially planned trajectory for each robot $i$ does not account
for the motion of other robots or changing of the environment. Thus, online
replanning is necessary during the implementation. Two distinct cases are
considered for online replanning, one is based on local communication between
robots and the other assumes no communication between robots. In the former
case, each robot has access to information (i.e., broadcast data from other
robots) within its sensing region while the latter case does not.

###  III-A Local communication case

In this section, we consider that each robot can identify conflicts within its
sensing region using the information broadcast by other robots. Before
proceeding, the definition of a conflict is needed.

Let $p_{i}(t)=[x_{i}(t),y_{i}(t)]^{T}$ be the position of robot $i$ at time
$t$. Denote by $p_{i}([t,t+\Delta_{i}])$ the local trajectory of robot $i$,
where $\Delta_{i}=\min_{t^{\prime}\geq
t}\\{p_{i}(t^{\prime})\notin\mathcal{B}_{i}(p_{i}(t),R_{i})\\}$. We say there
is a _conflict_ between robot $i$ and $j$ at time $t$ if there exists a region
$X_{k}\in\Phi$ such that $p_{i}([t,t+\Delta_{i}])\cap X_{k}\neq\emptyset$ and
$p_{j}([t,t+\Delta_{j}])\cap X_{k}\neq\emptyset$. This means that the local
trajectories of robots $i$ and $j$ pass through the same region $X_{k}$.

Using the local trajectory information $p_{j}([t,t+\Delta_{j}])$ broadcast by
the neighboring robots $j\in\mathcal{N}_{i}(t):=\\{j:\|p_{i}(t)-p_{j}(t)\|\leq
R_{i}\\}$, each robot $i$ can detect conflicts. Once conflicts are detected,
online replanning is conducted to ensure that conflicts are avoided and the
LTL task is satisfied. The online replanning procedure consists of a local and
a global trajectory generation algorithms, which are built upon our previous
work [16].

The local trajectory generation algorithm is detailed in Algorithm 1. Whenever
conflicts are detected among robots, a priority hierarchy is first created to
coordinate the planning order. Subsequently, a sampling-based algorithm is
employed to create a local collision-free trajectory.

1:$\xi_{i}$, $\mathcal{B}_{i}$,
$\mathcal{P}_{i},V_{\mathcal{P}_{i}},\mathbb{O},\mathbb{O}_{conf}$

2:A local transition system $\mathcal{T}_{i}^{L}$ and a leaf node
$\xi_{i}^{f}$.

3:Initialize
$\mathcal{T}_{i}^{L}=(S_{i}^{L},S_{i,0}^{L},\mathcal{U}_{i},AP,\rightarrow_{i}^{L},L)$
and $\xi_{i}^{f}=\emptyset$, where $S_{i}^{L}=S_{i,0}^{L}=\xi_{i}$ and
$\rightarrow_{i}^{L}=\emptyset$.

4:for $k=1,\ldots,N_{i}^{\max}$ do,

5: $\xi_{s}\leftarrow\textit{generateSample}(SA_{i})$,

6: $\xi_{n}\leftarrow\textit{nearest}(S_{i}^{L},\xi_{s})$,

7: Solve the optimization program $\mathcal{P}(\xi_{n},\xi_{s},\tau_{s})$,
which returns $(\xi_{r},u_{i}^{*})$,

8: $B_{i}(\xi_{r})\leftarrow\textit{trackBuchiStates}(\mathcal{B}_{i})$,

9: if $B_{i}(\xi_{r})\neq\emptyset\wedge
V_{\mathcal{P}_{i}}(\xi_{r},B_{i}(\xi_{r}))<\infty$ then,

10: if
$\operatorname{proj}_{2}\left(\left[\xi_{n},\xi_{r}\right]\right)isObstaclesFree(\mathbb{O},\mathbb{O}_{conf})\wedge
safeMotion(\xi_{r},R_{safe})$ then,

11: $S_{i}^{L}\leftarrow
S_{i}^{L}\cup\\{\xi_{r}\\};\rightarrow_{i}^{L}=\rightarrow_{i}^{L}\cup\\{\xi_{n}\xrightarrow[]{u_{i}^{*}}\xi_{r}\\}$,

12: end if

13: end if

14: if $\texttt{proj}_{2}(\xi_{r})\notin\mathcal{B}(p_{i},R_{i})$, then

15: $k=N_{i}^{\max}+1$,

16: $\xi_{i}^{f}\leftarrow\xi_{r}$,

17: end if

18:end for

Algorithm 1 localTrajectoryGeneration

Algorithm 1 starts with randomly sampling the expanded sensing region of robot
$i$. Then by constructing a local transition system, a collision-free
trajectory is synthesized that simultaneously guarantees the fulfillment of
the LTL task $\varphi_{i}$. It takes as input the current state
$\xi_{i}=(x_{i},y_{i},\theta_{i})$ of robot $i$, the offline constructed NBA
$\mathsf{B}_{i}$ and PBA $\mathcal{P}_{i}$, the offline computed potential
function $V_{\mathcal{P}_{i}}$ (the definition and computation can be found in
[16]), the set of known static obstacles $\mathbb{O}$, and the set of conflict
regions $\mathbb{O}_{conf}^{t}$ as input. Firstly, a local transition system
$\mathcal{T}_{i}^{L}$ is initialized (line 1). At each iteration, a new state
$\xi_{s}$ is taken randomly from the sampling area
$SA_{i}:=\left\\{(p,\theta):p\in\mathcal{B}\left(p_{i},R_{i}+\eta\right)\right\\}$
(line 3), where $\eta>0$ is an offline constant which ensures that one can
sample outside of the sensing region $\mathcal{B}\left(p_{i},R_{i}\right)$,
where $p_{i}$ is the current position of robot $i$. Then through the function
$\textit{nearest}\left(S_{i}^{L},\xi_{s}\right)$ an RRT primitive is applied,
which returns the nearest state to $\xi_{s}$ in $S_{i}^{L}$ (line 4). At this
stage an optimization problem
$\mathcal{P}\left(\xi_{n},\xi_{s},\tau_{s}\right)$ is solved, so as to find
the closest reachable state from the new sample $\xi_{s}$:

$\min_{u_{i}\in\mathbb{U}_{i}}\left\|\xi_{r}-\xi_{s}\right\|$ subject to:
$\displaystyle\xi_{i}(0)=\xi_{n},$
$\displaystyle\xi_{r}=\xi_{n}+\int_{0}^{\tau_{s}}F_{i}\left(\xi_{i}(s),u_{i}\right)ds,$
$\displaystyle u_{i}\in\mathcal{U}_{i},$

where $\tau_{s}$ represents the sampling time, while
$F_{i}\left(\xi_{i}(s),u_{i}\right)$ describes the robot’s dynamics (2) (line
5). Once $\xi_{r}$ is obtained, the corresponding subset of valid Büchi states
$B_{i}\left(\xi_{r}\right)$ is computed using algorithm trackBuchiState (given
in [22], Algorithm 1) (line 6). If both conditions
$B_{i}\left(\xi_{r}\right)\neq\emptyset$ and
$V_{\mathcal{P}_{i}}\left(\xi_{r},B_{i}\left(\xi_{r}\right)\right)<\infty$ are
satisfied (which ensures the existence of a path originating from $\xi_{r}$
that leads to a self-reachable accepting state of $\mathcal{P}_{i}$), then a
potential new state is considered. Such state $\xi_{r}$ is added into
$S_{i}^{L}$ and the corresponding transition relation
$\xi_{n}\stackrel{{\scriptstyle u_{i}^{*}}}{{\rightarrow}}\xi_{r}$ is added
into $\rightarrow_{i}^{L}$, if the path connecting $\xi_{n}$ with $\xi_{r}$ is
obstacles free and the motion is considered safe. To verify these
requirements, two new algorithms are designed in this work. The collision free
requirement is checked through the algorithm isObstaclesFree, which computes
the distance between the line segment
$\operatorname{proj}_{2}\left(\left[\xi_{n},\xi_{r}\right]\right)$ and the set
of obstacles $\mathbb{O}\cup\mathbb{O}_{conf}$. If the line segment
$\operatorname{proj}_{2}\left(\left[\xi_{n},\xi_{r}\right]\right)$ is
collision-free, we further check the safety of the motion using algorithm
$safeMotion(\xi_{r},R_{safe})$ (considering the base footprint of the robot),
where $R_{safe}$ is the safe distance that we specified a priori (lines 7-11).
The algorithm halts when the local sampling tree extends beyond the sensing
area, and the leaf node $\xi_{i}^{f}$ is subsequently returned (as determined
by the corresponding state $\xi_{r}$) (lines 12-15).

The global trajectory generation is similar to the initial trajectory
generation, which uses the leaf node $\xi_{i}^{f}$ (which is obtained by
Algorithm 1) as input of the LTL core & planner, and replans the sequence of
actions in order to accomplish the LTL task $\varphi_{i}$.

###  III-B Communication-free case

A communication-free scenario is also considered in pursuit of extending the
work to other real-world scenarios where reliable communications between
robots may not be available. A reactive collision avoidance algorithm is
designed in combination with the local planner.

Before proceeding, the concept of “trap state” is needed. _Trap states_ are
PBA states from which the Büchi acceptance condition cannot be fulfilled,
i.e., states that cannot reach accepting states that appear infinitely often.
For the PBA $\mathcal{P}_{i}$ of robot $i$, the set of all trap states is
denoted $\mathbb{G}_{i}$. An algorithm for computing trap states is
implemented in [21].

In this case, the online replanning is activated for robot $i$ whenever
obstacles (can be other robots and unknown static/moving obstacles) are
detected within its sensing region $\mathcal{B}_{i}(p_{i},R_{i})$. Let
$\mathbb{O}_{obs}^{t}$ be the set of moving obstacles detected at time $t$. In
order to avoid trap states $\mathbb{G}_{i}$ (thus guaranteeing task
feasibility), the set of known static obstacles $\mathbb{O}$, and the set of
moving obstacles $\mathbb{O}_{obs}^{t}$ (thus guaranteeing safety), we
consider a local model predictive controller:

$\displaystyle\min_{u_{i}}$
$\displaystyle\int_{t=0}^{T}(\xi_{i}(t)-X_{des})^{T}Q(\xi_{i}(t)-X_{des})+u_{i}^{T}Ru_{i}$
(3) $\displaystyle+(\xi(T)-X_{des})^{T}Q_{N}(\xi(T)-X_{des})\hskip 8.5359pt$
$\displaystyle+\int_{t=0}^{T}{w_{\mathbb{O}_{i}}\frac{1}{dist(\xi(t),\mathbb{O}_{obs}^{t}\cup\mathbb{O})}+w_{\mathbb{G}_{i}}\frac{1}{dist(\xi(t),\mathbb{G}_{i})}}$
subject to:
$\displaystyle\xi_{i}(t)=\xi_{i}(0)+\int_{0}^{t}F_{i}\left(\xi_{i}(s),u_{i}\right)ds,$
$\displaystyle u_{i}\in\mathcal{U}_{i},$ $\displaystyle\xi_{i}(T)\in X_{des}$
$\displaystyle\xi_{i}(0)=\xi_{i},$

where $T$ is the horizon and $\xi_{i}$ is the current state. The first 2 terms
of the cost function constitute a quadratic cost of both state and input with
the matrices $Q,R,Q_{N}$ being positive definite and $X_{des}\in\Phi$ denotes
the goal state that one can choose based on the current satisfying trajectory
(recall that the satisfying trajectory is discrete and have a prefix-suffix
structure). The last term of the cost function considers both the distance
between robot $i$ and the obstacles $\mathbb{O}_{obs}^{t}\cup\mathbb{O}$
(which is updated at each time step) and the distance between robot $i$ and
the trap states $\mathbb{G}_{i}$, where the distance function is defined as
$dist(x,A):=\inf_{y\in A}\\{||x-y\|\\}$ and
$w_{\mathbb{O}_{i}},w_{\mathbb{G}_{i}}$ are the corresponding weight
parameters. For instance if one prioritizes the satisfaction of the given LTL
task rather than not collide with obstacles, the weight $w_{\mathbb{G}_{i}}$
can be set higher than $w_{\mathbb{O}_{i}}$.

##  IV Human-in-the-loop control

In this section, we consider the human-in-the-loop context, where the robots
are assigned to complete complex tasks specified by an LTL formula, while a
human operator can take over the control of the robot from the on-board
autonomous controller. The robot is required to respect human inputs, but at
the same time react to undesired (possibly dangerous) human behaviors in order
to preserve safety and task satisfaction. This is useful for guiding the robot
through challenging assignments.

The proposed strategy is a MIC with the intention of extending existing work
[17, 18] to MRSs. The MIC, inspired by [17, 18], for robot $i$ is given by:

$u_{i}(t)\triangleq
u_{i}^{r}(t)+\kappa\left(r,\mathbb{O},\mathbb{G}_{i}\right)u^{h}(t),$ (4)

where $u_{i}^{r}(t)$ is a given autonomous controller, the function
$\kappa\left(\xi_{i},\mathbb{O},\mathbb{G}_{i}\right)\in[0,1]$ is a smooth
function to be designed, and $u^{h}(t)$ is the human input function, which is
uncontrollable and unknown by the robot.

The autonomous controller $u_{i}^{r}(t)$ can be a function that navigates the
robot from one region $X_{s}$ of the current discrete plan to the next one
$X_{g}$ while staying within the workspace $\mathcal{W}$ when there are no
conflicts/obstacles are detected. Otherwise, the local trajectory generation
algorithm or the model predictive controller developed in Section III is
implemented to get $u_{i}^{r}(t)$. In order to guarantee the task satisfaction
for all human inputs, the function
$\kappa\left(\xi_{i},\mathbb{O},\mathbb{G}_{i}\right)$ is designed as:

$\displaystyle\kappa\left(\xi_{i},\mathbb{O},\mathbb{G}_{i}\right)\triangleq$
$\displaystyle
G_{\mathrm{mix}}\cdot\frac{\rho\left(d_{o}-d_{s}\right)}{\rho\left(d_{o}-d_{s}\right)+\rho\left(\varepsilon+d_{s}-d_{t}\right)}$
$\displaystyle+(1-G_{\mathrm{mix}})\cdot\frac{\rho\left(d_{t}-d_{s}\right)}{\rho\left(d_{t}-d_{s}\right)+\rho\left(\varepsilon+d_{s}-d_{t}\right)}$

where $d_{t}\triangleq\min_{\pi\in\mathbb{G}_{i}}\|\xi_{i}-\pi\|$ is the
minimum distance between robot $i$ and any region within $\mathbb{G}_{i}$;
$d_{o}\triangleq\min_{\pi\in\mathbb{O}_{i}^{t}}\|\xi_{i}-\pi\|$ is the minimum
distance between the robot and any obstacle within $\mathbb{O}$;
$\rho(s)\triangleq$ $e^{-1/s}$ for $s>0$ and $\rho(s)\triangleq 0$ for $s\leq
0$, and $d_{s},\varepsilon>0$ are design parameters as the safety distance and
a small buffer. Moreover $G_{\mathrm{mix}}\in[0,1]$ represents a gain
parameter, in order to manage the trade-off between two aspects: preventing
trap states and obstacle avoidance.

##  V Experimental results

In this section, we present experimental studies to validate our results,
which is also the main contribution of this paper. The proposed strategies
hold potential for application in precision agriculture, enabling farmworkers
to collaborate effectively with robot teams in carrying out agronomic tasks,
such as harvesting or pruning in table-grape vineyards.

We use the Rosie HEBI mobile base (see Fig. 1 left), which is an
omnidirectional mobile platform with three omnidirectional wheels. Through the
use of the Qualisys motion capture system, the motion of the involved rigid
bodies is tracked in real time. Each HEBI Rosie mobile base includes an on-
board computer equipped with a proper ROS version, and autonomous control is
accomplished from the user PC using the ROS API in conjunction with a wireless
platform connection. The experiment workspace is an $5m\times 6m$ region as
shown in Fig. 1 right, which is abstracted into 30 $1m\times 1m$ squares. We
consider a group of 2 or 3 HEBI Rosie robots (Rosies 0, 1, and 2), and the
dynamics of each robot is given by (2). In addition, each Rosie is subject to
the inputs constraints $|v_{i}|\leq 0.35m/s$ and $|w_{i}|\leq 0.35rad/s$. The
sensing radius of each robot is $R_{i}=0.8m,\forall i$.

Figure 1: Rosie HEBI mobile base (left) and workspace discretization (right).

###  V-A Online replanning

In this section, we consider $N=2$ HEBI Rosie robots. The LTL specification
for each robot is given by

  * •

$\varphi_{0}=((\square\Diamond R8)\land(\square\Diamond R20))$,

  * •

$\varphi_{1}=(\square\Diamond R17)\land(\square\Diamond R21)$.

####  V-A1 Local communication case

Initially each robot synthesizes a satisfying trajectory using the LTL core &
planner [21]. During the online implementation, possible collisions are
detected, and then the local trajectory generation algorithm (Algorithm 1) is
executed taking into consideration the future trajectories of neighboring
robots (which is acquired through local communication). The real-time position
trajectories of the 2 Rosies are depicted in Fig. 2. During the experiment
time horizon $140s$, both robots successfully complete the surveillance tasks.
Conflicts are detected 4 times in total, and the local trajectory generation
algorithm is activated to resolve them every time. Note also that the input
constraints are satisfied at all time. A video demonstration of this
experiment can be found at https://www.youtube.com/watch?v=mKWpqvMrW9Y.

Figure 2: Local communication case: the real-time position trajectories of
Rosies 0 (green line) and 1 (purple line). The light green and purple regions
represent the target regions of Rosies 0 and 1, respectively.

####  V-A2 Communication-free case with human as moving obstacle

In this case, we additionally consider a human walking in the workspace
$\mathcal{W}$ to evaluate the effectiveness of the model predictive
controller.

Figure 3: Communication-free case: the real-time position trajectories of
Rosies 0 (green line), 1 (purple line), and human (black line).

Fig. 3 depicts the real-time position trajectories of the 2 Rosies and the
human, where the proposed model predictive controller (3) is applied for both
Rosies. Differently from the local communication case where replanning is
activated only if conflicts are detected, in the communication-free case, the
online replanning is activated whenever a robot detects obstacles in its
sensing region. As a result, the motion of each robot is affected not only by
other robots, but also by the human randomly walking through the workspace.

Replanning is conducted 14 times in total over the experiment time horizon
$120s$, and one can see that the model predictive controller (3) guarantees
the task satisfaction and collision avoidance (with other robots and the
moving human) for each robot. When compared to the local communication case,
the number of replanning is higher since the replanning process is activated
whenever obstacles are detected and the model predictive controller does not
take into account future trajectories of other robots. A video demonstration
of this experiment can be found at https://youtu.be/TYgfbrk7hDs.

###  V-B Human in-the-loop control

Figure 4: Human in-the-loop control.

This section aims to evaluate the MIC explained in Section V in the human in-
the-loop context, where a human may take control of one or more robots during
the motion. We consider $N=3$ HEBI Rosie robots and the LTL specification for
each robot is given by:

  * •

$\varphi_{0}=(\square\Diamond R8)\land(\square\Diamond R20)$,

  * •

$\varphi_{1}=(\square\Diamond R22)\land(\square\Diamond R28)$,

  * •

$\varphi_{2}=(\square\Diamond R10)\land(\square\Diamond R11)$.

Each robot can be controlled by a human with control inputs executed through a
joystick equipped with bluetooth, see Fig. 4. The human operator, on one hand,
influences robot autonomy when evaluating the performance of the proposed MIC,
where the human inputs and autonomous inputs are fused, as detailed in Section
V. On the other hand, the human operator is also walking through the workspace
as a moving obstacle. For the first $60s$, all robots are not affected by the
human obstacle and no joystick inputs are applied. After that, the human
starts to control Rosie 1 and walk through the workspace, and the model
predictive controller (communication-free case) is applied to deal with
potential conflicts for each robot. Additionally, some dangerous human
behaviours are tested, including the human attempting to guide the robot
towards itself. The real-time position trajectories of the 3 Rosies and the
human are depicted in Fig. 5. One can see that the MIC (4) prevents undesired
actions and ensures safety. A video demonstration of this experiment can be
found at https://youtu.be/2hWe5Wu52Bg.

Figure 5:  Human-in-the-loop: the real-time position trajectories of Rosies 0
(green line), 1 (purple line), 2 (red line), and human (black line), where
Rosie 1 is controlled by the human operator. The light green, purple, and red
regions represent the target regions of Rosies 0, 1, and 2, respectively.

##  VI Conclusions

The work focuses on the design of efficient planning and control algorithms
for MRSs that are subject to LTL specifications. Due to the practical
assumption of local sensing and dynamic environments, online replanning is
needed for each robot to guarantee safety and task satisfaction. Two distinct
cases are considered for online replanning. The former one assumes local
communication between robots and a local trajectory generation algorithm is
proposed to resolve conflicts. The latter one assumes no communication between
robots and a model predictive controller is designed to deal with penitential
collisions. Finally, the human-in-the-loop context is considered a MIC is
adopted to prevent unsafe human behavior.

## References

  * [1] C. Baier and J.-P. Katoen, _Principles of Model Checking_. MIT press, 2008.
  * [2] G. E. Fainekos, A. Girard, H. Kress-Gazit, and G. J. Pappas, “Temporal logic motion planning for dynamic robots,” _Automatica_ , vol. 45, no. 2, pp. 343–352, 2009.
  * [3] C. Belta, A. Bicchi, M. Egerstedt, E. Frazzoli, E. Klavins, and G. J. Pappas, “Symbolic planning and control of robot motion [grand challenges of robotics],” _IEEE Robotics & Automation Magazine_, vol. 14, no. 1, pp. 61–70, 2007.
  * [4] M. Kloetzer and C. Belta, “A fully automated framework for control of linear systems from temporal logic specifications,” _IEEE Transactions on Automatic Control_ , vol. 53, no. 1, pp. 287–297, 2008.
  * [5] T. Wongpiromsarn, U. Topcu, and R. M. Murray, “Receding horizon temporal logic planning,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 11, pp. 2817–2830, 2012.
  * [6] A. Ulusoy, S. L. Smith, X. C. Ding, C. Belta, and D. Rus, “Optimality and robustness in multi-robot path planning with temporal logic constraints,” _The International Journal of Robotics Research_ , vol. 32, no. 8, pp. 889–911, 2013.
  * [7] Y. E. Sahin, P. Nilsson, and N. Ozay, “Multirobot coordination with counting temporal logics,” _IEEE Transactions on Robotics_ , 2019.
  * [8] Y. Kantaros and M. M. Zavlanos, “STyLuS*: A temporal logic optimal control synthesis algorithm for large-scale multi-robot systems,” _The International Journal of Robotics Research_ , vol. 39, no. 7, pp. 812–836, 2020\.
  * [9] J. Alonso-Mora, J. A. DeCastro, V. Raman, D. Rus, and H. Kress-Gazit, “Reactive mission and motion planning with deadlock resolution avoiding dynamic obstacles,” _Autonomous Robots_ , vol. 42, no. 4, pp. 801–824, 2018\.
  * [10] M. Guo and D. V. Dimarogonas, “Multi-agent plan reconfiguration under local LTL specifications,” _The International Journal of Robotics Research_ , vol. 34, no. 2, pp. 218–235, 2015.
  * [11] M. Guo and M. M. Zavlanos, “Probabilistic motion planning under temporal tasks and soft constraints,” _IEEE Transactions on Automatic Control_ , vol. 63, no. 12, pp. 4051–4066, 2018.
  * [12] C. Belta, B. Yordanov, and E. A. Gol, _Formal methods for discrete-time dynamical systems_. Springer, 2017, vol. 15.
  * [13] T. Fong, I. Nourbakhsh, and K. Dautenhahn, “A survey of socially interactive robots,” _Robotics and autonomous systems_ , vol. 42, no. 3-4, pp. 143–166, 2003.
  * [14] M. Guo, K. H. Johansson, and D. V. Dimarogonas, “Revising motion planning under linear temporal logic specifications in partially known workspaces,” in _2013 IEEE international conference on robotics and automation_. IEEE, 2013, pp. 5025–5032.
  * [15] M. Lahijanian, M. R. Maly, D. Fried, L. E. Kavraki, H. Kress-Gazit, and M. Y. Vardi, “Iterative temporal planning in uncertain environments with partial satisfaction guarantees,” _IEEE Transactions on Robotics_ , vol. 32, no. 3, pp. 583–599, 2016.
  * [16] P. Yu and D. V. Dimarogonas, “Distributed motion coordination for multirobot systems under LTL specifications,” _IEEE Transactions on Robotics_ , vol. 38, no. 2, pp. 1047–1062, 2021.
  * [17] S. G. Loizou and V. Kumar, “Mixed initiative control of autonomous vehicles,” in _Proceedings 2007 IEEE International Conference on Robotics and Automation_. IEEE, 2007, pp. 1431–1436.
  * [18] M. Guo, S. Andersson, and D. V. Dimarogonas, “Human-in-the-loop mixed-initiative control under temporal tasks,” in _2018 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2018, pp. 6395–6400.
  * [19] J. R. Büchi, “On a decision method in restricted second order arithmetic,” in _The Collected Works of J. Richard Büchi_. Springer, 1990, pp. 425–435.
  * [20] P. Gastin and D. Oddoux, “Fast LTL to Büchi automata translation,” in _International Conference on Computer Aided Verification_. Springer, 2001, pp. 53–65.
  * [21] R. Baran, X. Tan, P. Varnai, P. Yu, S. Ahlberg, M. Guo, W. S. Cortez, and D. V. Dimarogonas, “A ros package for human-in-the-loop planning and control under linear temporal logic tasks,” in _2021 IEEE 17th International Conference on Automation Science and Engineering (CASE)_. IEEE, 2021, pp. 2182–2187.
  * [22] C. I. Vasile and C. Belta, “Sampling-based temporal logic path planning,” in _2013 IEEE/RSJ International Conference on Intelligent Robots and Systems_ , 2013, pp. 4817–4822.


# A new complemented subspace for the Lorentz sequence spaces, with an
application to its lattice of closed ideals

Ben Wallis  Division of Math/Science/Business, Kishwaukee College, Malta, IL
60150, United States<EMAIL_ADDRESS>
###### Abstract.

We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a
1-complemented subspace $Y$ distinct from $\ell_{p}$ and containing no
isomorph of $d(\textbf{w},p)$. In the general case, this is only the second
nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give
an explicit representation of $Y$ in the special case
$\textbf{w}=(n^{-\theta})_{n=1}^{\infty}$ ($0<\theta<1$) as the $\ell_{p}$-sum
of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find
a sixth distinct element in the lattice of closed ideals of
$\mathcal{L}(d(\textbf{w},p))$, of which only five were previously known in
the general case.

###### Key words and phrases:

Lorentz sequence spaces, complemented subspaces, lattice of closed ideals.

###### 2010 Mathematics Subject Classification:

Primary 46B20; Secondary 46B45, 47L20.

##  1\. Introduction

Little is known about the complemented subspace structure of Lorentz sequence
spaces $d(\textbf{w},p)$. Until recently, the only nontrivial complemented
subspace discussed in the literature was $\ell_{p}$ ([ACL73]). Then, in [Wa20]
it was shown that for certain weights w (see Theorem 2.2 below), the space
$d(\textbf{w},p)$ contains a 1-complemented subspace isomorphic to
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{p}$. Up to now, these were the
only nontrivial complemented subspaces known to exist.

In this short note we show that each Lorentz sequence space admits a
1-complemented subspace $Y$ distinct from $\ell_{p}$ (§2). We also give an
explicit representation of $Y$ for the case
$\textbf{w}=(n^{-\theta})_{n=1}^{\infty}$ ($0<\theta<1$), as the
$\ell_{p}$-sum of finite-dimensional copies of $d(\textbf{w},p)$ (§3).
Finally, as an application we find a sixth distinct element in the lattice of
closed ideals in the operator algebra $\mathcal{L}(d(\textbf{w},p))$, where
only five were previously known in the general case (§4).

Let’s set up the main notation we need to use. We begin by fixing
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$. Denote by $\Pi$ the set of all
permutations of $\mathbb{N}$, and denote by $\mathbb{W}$ the set of all
sequences $\textbf{w}=(w_{n})_{n=1}^{\infty}\in c_{0}\setminus\ell_{1}$
satisfying

$1=w_{1}\geqslant w_{2}\geqslant w_{3}\geqslant\cdots>0.$

Fix $1\leqslant p<\infty$ and $\textbf{w}\in\mathbb{W}$. For each
$(a_{n})_{n=1}^{\infty}\in\mathbb{K}^{\mathbb{N}}$ we set

$\left\|(a_{n})_{n=1}^{\infty}\right\|_{d(\textbf{w},p)}:=\sup_{\pi\in\Pi}\left(\sum_{n=1}^{\infty}|a_{\pi(n)}|^{p}w_{n}\right)^{1/p},$

and let $d(\textbf{w},p)$ denote the linear space of all
$(a_{n})_{n=1}^{\infty}\in\mathbb{K}^{\mathbb{N}}$ with
$\|(a_{n})_{n=1}^{\infty}\|_{d(\textbf{w},p)}<\infty$ endowed with the norm
$\|\cdot\|_{d(\textbf{w},p)}$, called a Lorentz sequence space. Recall that if
$(a_{n})_{n=1}^{\infty}\in c_{0}$ then there exists a “decreasing
rearrangement” $(\hat{a}_{n})_{n=1}^{\infty}$ of $(|a_{n}|)_{n=1}^{\infty}$.
In this case, the rearrangement inequality gives us

$\left\|(a_{n})_{n=1}^{\infty}\right\|_{d(\textbf{w},p)}=\left(\sum_{n=1}^{\infty}\hat{a}_{n}^{p}w_{n}\right)^{1/p}\;\;\;\text{
for all }(a_{n})_{n=1}^{\infty}\in c_{0}.$

Since $d(\textbf{w},p)\subset c_{0}$ as linear spaces (although not as normed
spaces), this represents an alternative formulation of the Lorentz sequence
space norm.

For each $i,k\in\mathbb{N}$ we define

$W_{k}:=\sum_{n=1}^{k}w_{n}\;\;\text{ and
}\;\;w_{i}^{(k)}:=\frac{1}{W_{k}}\sum_{n=(i-1)k+1}^{ik}w_{n}$

and $\textbf{w}^{(k)}:=(w_{i}^{(k)})_{i=1}^{\infty}$. It is readily apparent
that $\textbf{w}^{(k)}\in\mathbb{W}$. When $p$ is clear from context, we also
set

$d_{i}^{(k)}:=\frac{1}{W_{k}^{1/p}}\sum_{n=(i-1)k+1}^{ik}d_{n}.$

It’s routine to verify that $(d_{i}^{(k)})_{i=1}^{\infty}$ is a normalized
basic sequence isometrically equivalent to the $d(\textbf{w}^{(k)},p)$ basis.
If necessary, we may sometimes abuse this notation; for instance, if
$(j_{k})_{k=1}^{\infty}$ is a sequence in $\mathbb{N}$, then we could write
$((d_{i}^{(j_{k})})_{i=1}^{k})_{k=1}^{\infty}$ for appropriately-translated
successive normalized constant-coefficient blocks of lengths $j_{k}$.

Our main tool for finding complemented subspaces of $d(\textbf{w},p)$ is the
fact that every constant-coefficient block basic sequence of a symmetric basis
spans a 1-complemented subspace (cf. e.g. [LT77, Proposition 3.a.4]). We will
use this well-known fact freely and without further reference.

##  2\. Lorentz sequence spaces contain at least two nontrivial complemented
subspaces

The first discovery of a nontrivial complemented subspace in $d(\textbf{w},p)$
came almost half a century ago, with the following result.

######  Theorem 2.1 ([ACL73, Lemma 1]).

Fix $1\leqslant p<\infty$ and $\textbf{\emph{w}}\in\mathbb{W}$, and let

$x_{i}=\sum_{n=p_{i}}^{p_{i+1}-1}a_{n}d_{n},\;\;\;i\in\mathbb{N},$

form a seminormalized block basic sequence in $d(\textbf{\emph{w}},p)$. If
$a_{n}\to 0$ then $(x_{i})_{i=1}^{\infty}$ admits a subsequence equivalent to
$\ell_{p}$ and complemented in $d(\textbf{\emph{w}},p)$.

By taking sufficiently long constant-coefficient blocks, it follows that
$d(\textbf{w},p)$ contains a 1-complemented copy of $\ell_{p}$. Much later was
shown the following.

######  Theorem 2.2 ([Wa20, Theorem 4.3]).

Let $1\leqslant p<\infty$ and
$\textbf{\emph{w}}=(w_{n})_{n=1}^{\infty}\in\mathbb{W}$. If

$\inf_{k\in\mathbb{N}}\frac{\sum_{n=1}^{2k}w_{n}}{\sum_{n=1}^{k}w_{n}}=1$

then $d(\textbf{\emph{w}},p)$ admits a 1-complemented subspace spanned by
constant-coefficient blocks and isomorphic to
$(\bigoplus_{n=1}^{\infty}\ell_{\infty}^{n})_{p}$.

Thanks in large part to the ideas of William B. Johnson, our main result in
this section is to generalize this to all Lorentz sequence spaces, as follows.

######  Theorem 2.3.

Let $1\leqslant p<\infty$ and $\textbf{\emph{w}}\in\mathbb{W}$. Then there
exists an increasing sequence
$(N_{k})_{k=1}^{\infty}\in\mathbb{N}^{\mathbb{N}}$ such that
$((d_{i}^{(k)})_{i=1}^{N_{k}})_{k=1}^{\infty}$ spans a 1-complemented subspace
$Y$ which contains no isomorph of $d(\textbf{\emph{w}},p)$ and which is not
isomorphic to $\ell_{p}$.

To prove it, we need a few preliminaries.

######  Lemma 2.4.

If $1<p<\infty$ then every complemented subspace of $L_{p}[0,1]$ with a
subsymmetric basis $(x_{n})_{n=1}^{\infty}$ is isomorphic to either $\ell_{p}$
or $\ell_{2}$.

###### Proof.

The case $p=2$ is trivial since every complemented subspace of $L_{2}[0,1]$ is
isomorphic to $\ell_{2}$. For the case $p>2$ recall from [KP62, Corollary 6]
that every seminormalized basic sequence in $L_{p}[0,1]$, $p\in(2,\infty)$,
admits a subsequence equivalent to $\ell_{p}$ or $\ell_{2}$, and so since
$(x_{n})_{n=1}^{\infty}$ is also subsymmetric then it is in fact equivalent to
$\ell_{p}$ or $\ell_{2}$. In case $1<p<2$, since $(x_{n})_{n=1}^{\infty}$ is
complemented in $L_{p}[0,1]$, its corresponding sequence of biorthogonal
functionals $(x_{n}^{*})_{n=1}^{\infty}$ is contained in
$L_{p^{\prime}}[0,1]$, where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Since
$p^{\prime}>2$, a subsequence of $(x_{n}^{*})_{n=1}^{\infty}$ is equivalent to
$\ell_{p^{\prime}}$ or $\ell_{2}$, whence by subsymmetry
$(x_{n})_{n=1}^{\infty}$ is equivalent to $\ell_{p}$ or $\ell_{2}$. ∎

######  Lemma 2.5.

Let $X$ be a Banach space whose canonical isometric copy in $X^{**}$ is
complemented. Then for any free ultrafilter $\mathcal{U}$ on $\mathbb{N}$, the
canonical copy of $X$ in $X^{\mathcal{U}}$ is complemented in
$X^{\mathcal{U}}$.

###### Proof.

Let $q:X\to X^{**}$ denote the canonical embedding, and define the norm-1
linear operator $V:\ell_{\infty}(X)\to X^{**}$ by the rule

$V(x_{n})_{n=1}^{\infty}=\underset{\mathcal{U}}{\text{weak*-lim}}\,qx_{n},$

which exists by the weak*-compactness of $B_{X^{**}}$ together with the fact
that if $K$ is a compact Hausdorff space then for each
$(k_{n})_{n=1}^{\infty}\in K^{\mathbb{N}}$ the (unique) limit
$\lim_{\mathcal{U}}k_{n}$ exists in $K$. Note that if
$\lim_{\mathcal{U}}x_{n}=0$ then $V(x_{n})_{n=1}^{\infty}=0$ and so $V$
induces an operator $\widehat{V}:X^{\mathcal{U}}\to X^{**}$ which agrees with
$V$ along the diagonal. In particular, $\widehat{V}$ sends the canonical copy
of $X$ in $X^{\mathcal{U}}$ isomorphically to the canonical copy of $X$ in
$X^{**}$. ∎

######  Theorem 2.6.

Fix $1\leqslant p<\infty$, and let $(x_{n})_{n=1}^{\infty}$ be a basis for a
Banach space $X$ whose canonical copy in $X^{**}$ is complemented. If the
finite-dimensional spaces $[x_{n}]_{n=1}^{N}$, $N\in\mathbb{N}$, are uniformly
complemented in $L_{p}(\mu)$ for some measure $\mu$, then $X$ is complemented
in $L_{p}[0,1]$.

###### Proof.

Let $X_{N}=[x_{n}]_{n=1}^{N}$ and $P_{N}:X\to X_{N}$ the projection onto
$X_{N}$. By uniform complementedness of $X_{N}$, we can find uniformly bounded
linear operators $A_{N}:X_{N}\to L_{p}(\mu)$ and $B_{N}:L_{p}(\mu)\to X_{N}$
such that $B_{N}A_{N}$ is the identity on $X_{N}$. Let $\mathcal{U}$ be any
free ultrafilter on $\mathbb{N}$. Define the bounded linear operators $A:X\to
L_{p}(\mu)^{\mathcal{U}}$ by the rule $Ax=(A_{N}P_{N}x)_{\mathcal{U}}$, and
$B:L_{p}(\mu)^{\mathcal{U}}\to X^{\mathcal{U}}$ by
$B(y_{N})_{\mathcal{U}}=(B_{N}y_{N})_{\mathcal{U}}$. Let
$x\in\text{span}(x_{n})_{n=1}^{\infty}$ so that, for some $k\in\mathbb{N}$,

$BAx=(P_{1}x,\cdots,P_{k}x,x,x,\cdots)_{\mathcal{U}}=x^{\mathcal{U}}.$

By continuity, $BA$ is the canonical injection of $X$ into $X^{\mathcal{U}}$.
Since this is complemented by Lemma 2.5, we have the identity on $X$ factoring
through $L_{p}(\mu)^{\mathcal{U}}$.

It was proved in [He80, Theorem 3.3] that ultrapowers preserve $L_{p}$ lattice
structure, and in particular $L_{p}(\mu)^{\mathcal{U}}$ is isomorphic to
$L_{p}(\nu)$ for some measure $\nu$. Although $L_{p}(\nu)$ itself is
nonseparable, we could pass to the closed sublattice generated by $AX$ to find
a space isomorphic to a separable $L_{p}$ containing a complemented copy of
$X$. Due mostly to a famous result of Lacey and Wojtaszczyk, it’s known that
separable and infinite-dimensional $L_{p}$ spaces are isomorphic to either
$\ell_{p}$ or $L_{p}[0,1]$ ([JL01, §4, p15]). This means an isomorph of $X$ is
complemented in $L_{p}[0,1]$. ∎

An immediate corollary to Lemma 2.4 and Theorem 2.6 is as follows.

######  Corollary 2.7.

Let $1<p<\infty$ and $\textbf{\emph{w}}\in\mathbb{W}$. Then no $L_{p}(\mu)$
space contains uniformly complemented copies of $[d_{n}]_{n=1}^{N}$,
$N\in\mathbb{N}$.

Now we’re ready to prove the main result of this section.

###### Proof of Theorem 2.3.

Fix $k\in\mathbb{N}$, and note that $(d_{i}^{(k)})_{k=1}^{\infty}$ is
isometric to the $d(\textbf{w}^{(k)},p)$ basis. Consider the case where $p=1$.
Then we can choose the $N_{k}$’s large enough that each
$(d_{i}^{(k)})_{i=1}^{N_{k}}$ fails to be $k$-equivalent to
$\ell_{1}^{N_{k}}$, and hence $((d_{i}^{(k)})_{i=1}^{N_{k}})_{k=1}^{\infty}$
fails to be equivalent to $\ell_{1}$. As $\ell_{1}$ has a unique unconditional
basis by a result of Lindenstrauss and Pełczyński, it follows that $Y$ is not
isomorphic to $\ell_{1}$.

Next, consider the case where $1<p<\infty$. By Corollary 2.7 we can select
$N_{k}$’s large enough that $[d_{i}^{(k)}]_{i=1}^{N_{k}}$ fails to be
$k$-complemented in $\ell_{p}$. As $[d_{i}^{(k)}]_{i=1}^{N_{k}}$’s are all
1-complemented in $Y$, that means $Y$ is not isomorphic to $\ell_{p}$.

It remains to show that $Y$ contains no isomorph of $d(\textbf{w},p)$. Suppose
towards a contradiction that it does. As $(d_{n})_{n=1}^{\infty}$ is weakly
null (cf. e.g. [ACL73, Proposition 1]) we can use the gliding hump method
together with symmetry to find a normalized block sequence of
$((d_{i}^{(k)})_{i=1}^{N_{k}})_{k=1}^{\infty}$ equivalent to
$(d_{n})_{n=1}^{\infty}$. However, every such block sequence is also a block
sequence w.r.t. $(d_{n})_{n=1}^{\infty}$ with coefficients tending to zero. By
Theorem 2.1 it follows that $(d_{n})_{n=1}^{\infty}$ admits a subsequence
equivalent to $\ell_{p}$, which is impossible. ∎

##  3\. A special case

In this section we show that when $\textbf{w}=(n^{-\theta})_{n=1}^{\infty}$
for some fixed $0<\theta<1$, the space $Y$ described in Theorem 2.3 can be
chosen to be isomorphic to the space

$Y_{\textbf{w},p}:=\left(\bigoplus_{N=1}^{\infty}D_{N}\right)_{p},$

where $D_{N}:=[d_{n}]_{n=1}^{N}$ for each $N\in\mathbb{N}$. As usual, we
require some preliminaries.

######  Lemma 3.1.

Let $0<\theta<1$ and $j,k\in\mathbb{N}$. Then

$((j+1)/k+1)^{1-\theta}-((j+1)/k)^{1-\theta}\leqslant\frac{\sum_{n=j+1}^{j+k}n^{-\theta}}{\sum_{n=1}^{k}n^{-\theta}}\leqslant\frac{(j/k+1)^{1-\theta}-(j/k)^{1-\theta}}{2^{1-\theta}-1}.$

###### Proof.

Observe that the map

$f(t)=(1+1/t)^{1-\theta}-(1/t)^{1-\theta}$

is increasing on $[1,\infty)$, and hence has a minimum $f(1)=2^{1-\theta}-1$.
Hence,

$\displaystyle((j+1)/k+1)^{1-\theta}-((j+1)/k)^{1-\theta}$
$\displaystyle\leqslant\frac{(j+k+1)^{1-\theta}-(j+1)^{1-\theta}}{k^{1-\theta}-\theta}$
$\displaystyle=\frac{\int_{j+1}^{j+k+1}t^{-\theta}\;dt}{1+\int_{1}^{k}t^{-\theta}\;dt}$
$\displaystyle\leqslant\frac{\sum_{n=j+1}^{j+k}n^{-\theta}}{\sum_{n=1}^{k}n^{-\theta}}$
$\displaystyle\leqslant\frac{\int_{j}^{j+k}t^{-\theta}\;dt}{\int_{1}^{k+1}t^{-\theta}\;dt}$
$\displaystyle=\frac{(j+k)^{1-\theta}-j^{1-\theta}}{(k+1)^{1-\theta}-1}$
$\displaystyle=\frac{(j/k+1)^{1-\theta}-(j/k)^{1-\theta}}{(1+1/k)^{1-\theta}-(1/k)^{1-\theta}}$
$\displaystyle\leqslant\frac{(j/k+1)^{1-\theta}-(j/k)^{1-\theta}}{2^{1-\theta}-1}.$

∎

######  Lemma 3.2.

Let $0<\theta<1$ and
$\textbf{\emph{w}}=(w_{n})_{n=1}^{\infty}=(n^{-\theta})_{n=1}^{\infty}\in\mathbb{W}$.
Then

$\frac{1-\theta}{2}\cdot w_{i}\leqslant
w_{i}^{(k)}\leqslant\frac{2-2^{\theta}}{2^{1-\theta}-1}\cdot w_{i}\;\;\;\text{
for all }i,k\in\mathbb{N}.$

In particular, if $1\leqslant p<\infty$ then there is a constant
$C\in[1,\infty)$, depending only on $\theta$, such that

$(d_{n})_{n=1}^{\infty}\approx_{C}(d_{i}^{(k)})_{i=1}^{\infty}\;\;\;\text{ for
all }k\in\mathbb{N}.$

###### Proof.

We can assume $i,k\geqslant 2$. Observe that

$t\mapsto t-(t-1)^{1-\theta}\cdot t^{\theta}$

is decreasing on $[2,\infty)$ and hence has the maximum $2-2^{\theta}$. Also,
the function

$t\mapsto t-(t-1/2)^{1-\theta}\cdot t^{\theta}$

is decreasing on $[2,\infty)$ and hence has infimum

$\lim_{t\to\infty}\left(t-(t-1/2)^{1-\theta}\cdot
t^{\theta}\right)=\frac{1-\theta}{2}.$

Thus, by the above together with Lemma 3.1,

$\displaystyle\frac{1-\theta}{2}\cdot i^{-\theta}$
$\displaystyle\leqslant\left(i-(i-1/2)^{1-\theta}\cdot
i^{\theta}\right)i^{-\theta}$ $\displaystyle=i^{1-\theta}-(i-1/2)^{1/\theta}$
$\displaystyle\leqslant(i+1/k)^{1-\theta}-(i-1+1/k)^{1/\theta}$
$\displaystyle\leqslant\frac{\sum_{n=(i-1)k+1}^{ik}n^{-\theta}}{\sum_{n=1}^{k}n^{-\theta}}$
$\displaystyle\leqslant\frac{i^{1-\theta}-(i-1)^{1-\theta}}{2^{1-\theta}-1}$
$\displaystyle=\frac{i-(i-1)^{1-\theta}\cdot i^{\theta}}{2^{1-\theta}-1}\cdot
i^{-\theta}$ $\displaystyle\leqslant\frac{2-2^{\theta}}{2^{1-\theta}-1}\cdot
i^{-\theta}.$

∎

######  Remark 3.3.

Suppose $x=\sum_{n\in A}a_{n}d_{n}$ and $y=\sum_{n\in B}b_{n}d_{n}$ for finite
and disjoint sets $A,B\subset\mathbb{N}$, where $(a_{n})_{n\in A}$ and
$(b_{n})_{n\in B}$ are sequences of scalars. Then

$\|x+y\|^{p}\leqslant\|x\|^{p}+\|y\|^{p}.$

######  Lemma 3.4.

Let $(j_{k})_{k=1}^{\infty}$ be a sequence of positive integers, and for each
$k$ set

$J_{k}=j_{1}+2j_{2}+3j_{3}+\cdots+kj_{k}.$

Suppose that there are constants $A,B\in(0,\infty)$ such that

$w_{i}^{(j_{k})}\leqslant Aw_{i}$ (3.1)

and

$Bw_{i}\leqslant\frac{1}{W_{j_{k}}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}}w_{n}$
(3.2)

for all $i=1,\cdots,k$ and all $k\in\mathbb{N}$. Then
$((d_{i}^{(j_{k})})_{i=1}^{k})_{k=1}^{\infty}$ is equivalent to the canonical
$Y_{\textbf{\emph{w}},p}$ basis.

###### Proof.

Due to (3.1) we have
$(d_{i}^{(j_{k})})_{i=1}^{k}\lesssim_{A}d(\textbf{w},p)^{k}$. Now, using
Remark 3.3, for any finitely-supported scalar sequence
$((a_{i}^{(k)})_{i=1}^{k})_{k=1}^{\infty}$,

$\displaystyle\left\|\sum_{k=1}^{\infty}\sum_{i=1}^{k}a_{i}^{(k)}d_{i}^{(j_{k})}\right\|^{p}$
$\displaystyle\leqslant\sum_{k=1}^{\infty}\left\|\sum_{i=1}^{k}a_{i}^{(k)}d_{i}^{(j_{k})}\right\|^{p}$
$\displaystyle\leqslant
A^{p}\sum_{k=1}^{\infty}\left\|(a_{i}^{(k)})_{i=1}^{k}\right\|_{d(\textbf{w},p)}^{p}$
$\displaystyle=A^{p}\left\|((a_{i}^{(k)})_{i=1}^{k})_{k=1}^{\infty}\right\|_{Y_{\textbf{w},p}}^{p}.$

For the reverse inequality, let $(\hat{a}_{i}^{(k)})_{i=1}^{k}$ denote the
decreasing rearrangement of $(|a_{i}^{(k)}|)_{i=1}^{k}$. Then, applying (3.2),

$\displaystyle\left\|\sum_{k=1}^{\infty}\sum_{i=1}^{k}a_{i}^{(k)}d_{i}^{(j_{k})}\right\|^{p}$
$\displaystyle=\left\|\sum_{k=1}^{\infty}\sum_{i=1}^{k}\frac{a_{i}^{(k)}}{W_{j_{k}}^{1/p}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}}d_{n}\right\|^{p}$
$\displaystyle\geqslant\sum_{k=1}^{\infty}\sum_{i=1}^{k}\frac{\hat{a}_{i}^{(k)p}}{W_{j_{k}}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}}w_{n}$
$\displaystyle\geqslant
B\sum_{k=1}^{\infty}\sum_{i=1}^{k}\hat{a}_{i}^{(k)p}w_{i}$
$\displaystyle=B\left\|((a_{i}^{(k)})_{i=1}^{k})_{k=1}^{\infty}\right\|_{Y_{\textbf{w},p}}^{p}.$

∎

######  Theorem 3.5.

Let $(j_{k})_{k=1}^{\infty}$ and $(J_{k})_{k=1}^{\infty}$ be is as in Lemma
3.4. Suppose there is $M\in[1,\infty)$ such that

$\frac{J_{k-1}}{j_{k}}\leqslant M,\;\;\;\text{ for all }k=2,3,4,\cdots.$

Then $((d_{i}^{(j_{k})})_{i=1}^{k})_{k=1}^{\infty}$ is equivalent to the
canonical $Y_{\textbf{\emph{w}},p}$ basis.

###### Proof.

Due to Lemma 3.4, it suffices to show that (3.2) and (3.1) both hold. To do
this, fix an arbitrary $k\in\mathbb{N}$. We may assume without loss of
generality that $j_{k}\geqslant 2$. Now, by Lemma 3.1,

$\displaystyle\frac{1}{W_{j_{k}}}\sum_{n=J_{k-1}+(i-1)j_{k}+1}^{J_{k-1}+ij_{k}}w_{n}$
$\displaystyle\geqslant\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}+1\right)^{1-\theta}-\left(\frac{J_{k-1}+(i-1)j_{k}+1}{j_{k}}\right)^{1-\theta}$
$\displaystyle=\left(\frac{J_{k-1}}{j_{k}}+i+\frac{1}{j_{k}}\right)^{1-\theta}-\left(\frac{J_{k-1}}{j_{k}}+i-1+\frac{1}{j_{k}}\right)^{1-\theta}$
$\displaystyle\geqslant\left(\frac{J_{k-1}}{j_{k}}+i\right)^{1-\theta}-\left(\frac{J_{k-1}}{j_{k}}+i-1+\frac{1}{2}\right)^{1-\theta}$
$\displaystyle=i^{\theta}\left[\left(\frac{J_{k-1}}{j_{k}}+i\right)^{1-\theta}-\left(\frac{J_{k-1}}{j_{k}}+i-\frac{1}{2}\right)^{1-\theta}\right]w_{i}$

Applying the Mean Value Theorem to the function $x\mapsto(\phi+x)^{1-\theta}$,
$\phi\in[1,\infty)$, we can find $x_{\phi}\in(-1/2,0)$ such that

$\phi^{1-\theta}-(\phi-1/2)^{1-\theta}=\frac{(1-\theta)(\phi+x_{\phi})^{-\theta}}{2}\geqslant\frac{(1-\theta)\phi^{-\theta}}{2}.$

Hence, letting $\phi=J_{k-1}/j_{k}+i$, we have

$\displaystyle
i^{\theta}\left[\left(\frac{J_{k-1}}{j_{k}}+i\right)^{1-\theta}-\left(\frac{J_{k-1}}{j_{k}}+i-\frac{1}{2}\right)^{1-\theta}\right]$
$\displaystyle\geqslant
i^{\theta}\left[\frac{(1-\theta)(J_{k-1}/j_{k}+i)^{-\theta}}{2}\right]$
$\displaystyle=\frac{1-\theta}{2}\left(\frac{i}{J_{k-1}/j_{k}+i}\right)^{\theta}$
$\displaystyle\geqslant\frac{1-\theta}{2}\left(\frac{1}{M+1}\right)^{\theta}$

This proves (3.2), and (3.1) follows immediately from Lemma 3.2. ∎

Taking inductively $j_{1}=1$ and $j_{k+1}=J_{k}$, the following is now
immediate.

######  Corollary 3.6.

Let $1\leqslant p<\infty$, $0<\theta<1$, and
$\textbf{\emph{w}}=(w_{n})_{n=1}^{\infty}=(n^{-\theta})_{n=1}^{\infty}\in\mathbb{W}$.
Then $d(\textbf{\emph{w}},p)$ admits a 1-complemented subspace isomorphic to
$Y_{\textbf{\emph{w}},p}$.

##  4\. application to the lattice of closed ideals

In [KPSTT12] was shown (among other results) that the lattice of closed ideals
for the operator algebra $\mathcal{L}(d(\textbf{w},p))$ can be put into a
chain:

$\\{0\\}\subsetneq\mathcal{K}(d(\textbf{w},p))\subsetneq\mathcal{SS}(d(\textbf{w},p))\subsetneq\mathcal{S}_{d(\textbf{w},p)}(d(\textbf{w},p))\subsetneq\mathcal{L}(d(\textbf{w},p)).$

Here, $\mathcal{K}$ denotes the compact operators, $\mathcal{SS}$ the strictly
singular operators, and $\mathcal{S}_{d(\textbf{w},p)}$ the ideal of operators
which fail to be bounded below on any isomorph of $d(\textbf{w},p)$. While in
[Wa20, Corollary 2.7], for the special case where $1<p<2$ and
$\textbf{w}\in\mathbb{W}\cap\ell_{2/(2-p)}$, a chain of distinct closed ideals
with cardinality of the continuum were identified lying between
$\mathcal{K}(d(\textbf{w},p))$ and $\mathcal{SS}(d(\textbf{w},p)$, for the
general case, the only distinct elements known were those of the above chain.

For an operator $T$, let $\mathcal{J}_{T}$ denote the class of operators
factoring through $T$. If $Z$ is any Banach space, we then set
$\mathcal{J}_{Z}=\mathcal{J}_{Id_{Z}}$. By Theorem 4.3 below, we can extend
the chain above as follows:

$\\{0\\}\subsetneq\mathcal{K}(d(\textbf{w},p))\subsetneq\mathcal{SS}(d(\textbf{w},p))\subsetneq(\overline{\mathcal{J}_{\ell_{p}}}\vee\mathcal{SS})(d(\textbf{w},p))\\\
\subsetneq\mathcal{S}_{d(\textbf{w},p)}(d(\textbf{w},p))\subsetneq\mathcal{L}(d(\textbf{w},p)).$$$

Furthermore, by [KPSTT12, Corollary 3.2 and Theorem 5.3] together with the
fact that $d(\textbf{w},p)$ has the approximation property, any additional
distinct closed ideals in $\mathcal{L}(d(\textbf{w},p))$ must lie between
$\mathcal{K}(d(\textbf{w},p))$ and $\mathcal{SS}(d(\textbf{w},p))$, or else
between $(\overline{\mathcal{J}_{\ell_{p}}}\vee\mathcal{SS})(d(\textbf{w},p))$
and $\mathcal{S}_{d(\textbf{w},p)}(d(\textbf{w},p))$.

To prove Theorem 4.3, we need a couple of preliminary results.

######  Proposition 4.1.

Let $X$ and $Z$ be an infinite-dimensional Banach spaces such that
$Z^{2}\approx Z$, and $X$ fails to be isomorphic to a complemented subspace of
$Z$. Then $\overline{\mathcal{J}_{Z}}(X)$ is a proper ideal in
$\mathcal{L}(X)$. Furthermore, if $P\in\mathcal{L}(X)$ is a projection with
image isomorphic to $Z$, then

$\mathcal{J}_{P}(X)=\mathcal{J}_{Z}(X).$

###### Proof.

Since $Z^{2}\approx Z$, [KPSTT12, Lemma 2.2] guarantees that
$\mathcal{J}_{Z}(X)$ is an ideal in $\mathcal{L}(X)$. Suppose towards a
contradiction that that $Id_{X}\in\mathcal{J}_{Z}(X)$. Then $Id_{X}=AB$ for
operators $A\in\mathcal{L}(Z,X)$ and $B\in\mathcal{L}(X,Z)$. By [KPSTT12,
Lemma 2.1], $BX$ is complemented in $Z$ and isomorphic to $X$, which
contradicts our hypotheses. It follows that $\mathcal{J}_{Z}(X)$ is a proper
ideal in $\mathcal{L}(X)$. Recall that the closure of a proper ideal in a
unital Banach algebra is again proper; in particular,
$\overline{\mathcal{J}_{Z}}(X)$ is a proper ideal in $\mathcal{L}(X)$.

To prove the “furthermore” part, assume $A\in\mathcal{L}(Z,X)$ and
$B\in\mathcal{L}(X,Z)$. Let $Q:Z\to X$ be the canonical embedding so that
$PQ=Id_{Z}$ and hence $AB=APQB\in\mathcal{J}_{P}(X)$. It follows that
$\mathcal{J}_{Z}(X)\subseteq\mathcal{J}_{P}(X)$, and the reverse inclusion is
even more obvious. ∎

For the next result, $\mathcal{F}$ denotes the class of finite-rank operators
and $\mathcal{E}$ the class of inessential operators. Recall also that a basis
$\mathcal{B}$ is called semispreading whenever every subsequence of
$\mathcal{B}$ is dominated by $\mathcal{B}$ itself. In particular, the unit
vector basis of $\ell_{p}$ is semispreading.

######  Proposition 4.2 ([LLR04, Corollary 3.8]).

Let $Z$ be a Banach space with a semi-spreading basis $(z_{n})$, and let $X$
be a Banach space with basis $(x_{n})$ such that any seminormalized block
sequence of $(x_{n})$ contains a subsequence equivalent to $(z_{n})$ and
spanning a complemented subspace of $X$. Then

$\\{0\\}\subsetneq\overline{\mathcal{F}}(X)=\mathcal{K}(X)=\mathcal{SS}(X)=\mathcal{E}(X)\subsetneq\overline{\mathcal{J}_{Z}}(X),$

and any additional distinct closed ideals must lie between
$\overline{\mathcal{J}_{Z}}(X)$ and $\mathcal{L}(X)$.

In the proof of what follows, we use the fact that if $\mathcal{I}$ and
$\mathcal{J}$ are ideals in $\mathcal{L}(X)$, then
$\overline{\mathcal{I}}\vee\overline{\mathcal{J}}=\overline{\mathcal{I}+\mathcal{J}}$.

######  Theorem 4.3.

Fix $1\leqslant p<\infty$ and $\textbf{\emph{w}}\in\mathbb{W}$. Let $Y$ be as
in Theorem 2.3, and $P_{Y}\in\mathcal{L}(d(\textbf{\emph{w}},p))$ any
continuous linear projection onto $Y$. Then

$P_{Y}\in\mathcal{S}_{d(\textbf{\emph{w}},p)}(d(\textbf{\emph{w}},p))\setminus(\overline{\mathcal{J}_{\ell_{p}}}\vee\mathcal{SS})(d(\textbf{\emph{w}},p)).$

###### Proof.

Let $P_{\ell_{p}}\in\mathcal{L}(d(\textbf{w},p))$ be any projection onto an
isomorophic copy of $\ell_{p}$ spanned by basis vectors of $Y$. (Such a copy
exists by Theorem 2.1.) By Theorem 2.3, $Y$ contains no isomorph of
$d(\textbf{w},p)$ and hence
$P_{Y}\in\mathcal{S}_{d(\textbf{w},p)}(d(\textbf{w},p))$. Since
$\mathcal{S}_{d(\textbf{w},p)}(d(\textbf{w},p))$ is the unique maximal ideal
in $\mathcal{L}(d(\textbf{w},p))$, and
$\mathcal{J}_{P_{\ell_{p}}}(d(\textbf{w},p))=\mathcal{J}_{\ell_{p}}(d(\textbf{w},p))$
by Proposition 4.1, it’s sufficient to prove that
$P_{Y}\notin(\overline{\mathcal{J}_{P_{\ell_{p}}}}\vee\mathcal{SS})(d(\textbf{w},p)).$

Next, we claim that
$P_{Y}\in(\overline{\mathcal{J}_{P_{\ell_{p}}}}\vee\mathcal{SS})(d(\textbf{w},p))$
only if $Id_{Y}\in(\overline{\mathcal{J}_{\ell_{p}}}\vee\mathcal{SS})(Y)$. To
prove it, fix $\varepsilon>0$, and suppose there are
$A,B\in\mathcal{L}(d(\textbf{w},p))$ and $S\in\mathcal{SS}(d(\textbf{w},p))$
such that

$\left\|AP_{\ell_{p}}B+S-P_{Y}\right\|<\varepsilon.$

Let $J_{Y}:Y\to d(\textbf{w},p)$ be an embedding satisfying
$P_{Y}J_{Y}=J_{Y}$, or $P_{Y}J_{Y}=Id_{Y}$ when viewed as an operator in
$\mathcal{L}(Y)$. Composing $P_{Y}$ on the left and $J_{Y}$ on the right, we
have

$\left\|P_{Y}AP_{\ell_{p}}BJ_{Y}+P_{Y}SJ_{Y}-Id_{Y}\right\|_{\mathcal{L}(Y)}<\|P_{Y}\|\cdot\varepsilon\cdot\|J_{Y}\|.$

On the other hand, since $AP_{\ell_{p}}=A|_{Y}P_{\ell_{p}}$ and
$P_{\ell_{p}}=P_{\ell_{p}}P_{Y}$, we have

$P_{Y}AP_{\ell_{p}}BJ_{Y}=(P_{Y}A|_{Y})P_{\ell_{p}}(P_{Y}BJ_{Y})$

and hence

$\left\|(P_{Y}A|_{Y})P_{\ell_{p}}(P_{Y}BJ_{Y})+P_{Y}SJ_{Y}-Id_{Y}\right\|_{\mathcal{L}(Y)}<\|P_{Y}\|\cdot\varepsilon\cdot\|J_{Y}\|.$

Since $\mathcal{J}_{\ell_{p}}(Y)=\mathcal{J}_{P_{\ell_{p}}}(Y)$ by Proposition
4.1, where $P_{\ell_{p}}$ is likewise viewed as an operator in
$\mathcal{L}(Y)$, from the above together with the ideal property of
$\mathcal{SS}$, the claim follows.

Let $\mathcal{B}_{Y}=((d_{i}^{(k)})_{i=1}^{N_{k}})_{k=1}^{\infty}$ denote the
canonical basis of $Y$ from Theorem 2.3. Note that since $\mathcal{B}_{Y}$ is
made up of constant coefficient blocks of $(d_{n})$ of increasing length, any
seminormalized blocks of $\mathcal{B}_{Y}$ will contain a subsequence
equivalent to $\ell_{p}$ by Theorem 2.1. In fact, in [CL74, Lemma 15] this
result was refined to show that we can choose that subsequence to span a
complemented subspace of $d(\textbf{w},p)$, and hence of $Y$ itself. We can
therefore apply Theorem 4.2 to conclude that
$\mathcal{SS}(Y)\subset\overline{\mathcal{J}_{\ell_{p}}}(Y)$. Meanwhile, again
by Proposition 4.1, $\overline{\mathcal{J}_{\ell_{p}}}(Y)$ is a proper ideal
in $\mathcal{L}(Y)$, which means
$Id_{Y}\notin\overline{\mathcal{J}_{\ell_{p}}}(Y)$. Hence,
$P_{Y}\notin(\overline{\mathcal{J}_{P_{\ell_{p}}}}\vee\mathcal{SS})(d(\textbf{w},p))$
as desired. ∎

## References

  * [ACL73] Zvi Altshuler, Peter G. Casazza and Bor-Luh Lin. “On symmetric basic sequences in Lorentz sequence spaces,” Israel Journal of Mathematics 15:2 (1973), pp140–55.
  * [CL74] Peter G. Casazza and Bor-Luh Lin. “On symmetric basic sequences in lorentz sequence spaces II,” Israel Journal of Mathematics 17 (1974), pp191–218.
  * [He80] Stefan Heinrich. “Ultraproducts in Banach space theory,” Journal fúr die reine und angewandte Mathematik 313 (1980), pp72–104.
  * [JL01] William B. Johnson and Joram Lindenstrauss. “Basic concepts in the geometry of Banach spaces,” Handbook of the Geometry of Banach Spaces, Volume 1 (2001), William B. Johnson and Joram Lindenstrauss, eds., ISBN 0-444-82842-7.
  * [KPSTT12] Anna Kaminska, Alexey I. Popov, Eugeniu Spinu, Adi Tcaciuc, and Vladimir G. Troitsky. “Norm closed operator ideals in Lorentz sequence spaces,” Journal of Mathematical Analysis and Applications 389:1 (2012), pp247–260.
  * [KP62] Mikhail I. Kadec and Aleksander Pełczyński. “Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$,” Studia Mathematica 21 (1962), pp161–176.
  * [LLR04] Niels Jakob Laustsen, Richard J. Loy, and Charles J. Read. “The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces,” Journal of Functional Analysis 214 (2004), pp106–131.
  * [LT77] Joram Lindenstrauss and Lior Tzafriri. Classical Banach Spaces I, Sequence Spaces, Springer, Berlin (1977), ISBN 3-540-08072-4
  * [Wa20] Ben Wallis. “Closed ideals of operators acting on some families of sequence spaces,” Banach Journal of Mathematical Analysis 14:1 (2020), pp98–115.