∎ 11institutetext: Georgios Drakopoulos and Phivos Mylonas 22institutetext:
Department of Informatics, Ionian University, Greece
22email: {c16drak<EMAIL_ADDRESS>33institutetext: Spyros Sioutas
44institutetext: Computer Engineering and Informatics Department, University
of Patras, Greece
44email<EMAIL_ADDRESS>
# An Architecture For Cooperative Mobile Health Applications

Georgios Drakopoulos  Phivos Mylonas  Spyros Sioutas

(Received: date / Accepted: date)

###### Abstract

Mobile health applications are steadily gaining momentum in the modern world
given the omnipresence of various mobile or WiFi connections. Given that the
bandwidth of these connections increases over time, especially in conjunction
with advanced modulation and error-correction codes, whereas the latency
drops, the cooperation between mobile applications becomes gradually easier.
This translates to reduced computational burden and heat dissipation for each
isolated device at the expense of increased privacy risks. This chapter
presents a configurable and scalable edge computing architecture for
cooperative digital health mobile applications.

###### Keywords:

Digital health Edge computing Mobile computing Mobile applications Cooperative
applications Higher order statistics

###### MSC:

05C12 05C20 05C80 05C85

††journal: GeNeDis

##  1 Introduction

###

Mobile smart applications for monitoring human health or processing health-
related data are increasing lately at an almost geometric rate. This can be
attributed to a combination of social and technolgical factors. The
accumulated recent multidisciplinary research on biosignals and the quest for
improved biomarkers bore fruits in the form of advanced bisignal processing
algorithms. Smartphone applications are progressively becoming popular in all
age groups, albeit with a different rate for each such group and, moreover,
mobile subscribers tend to be more willing to provide sensitive health data
such are heart beat rate, blood pressure, or eye sight status to applications
for processing. Thus, not only technological but also financial factors favor
the development of digital health applications.

###

The primary contribution of this chapter is a set of guidelines towards a
cross-layer cooperative architecture for mobile health applications. The
principal motivation behind them are increased parallelism, and consequently
lower turnaround or wallclock time, additional redundancy, which translates to
higher reliability, and lower energy consumption. All these factors are
critical for mobile health applications.

###

The remaining of this chapter is structured as follows. Section 2 briefly
summarizes the recent scientific literature in the fields of edge computing,
mobile applications, mobile services, and digital health applications. Section
3 presents the proposed architecture. Finally, section 4 recapitulates the
main points of this chapter. The notation of this chapter is shown at table 1.

Table 1: Notation of this chapter. Symbol | Meaning
---|---
${\buildrel{{\triangle}}\over{=}}$ | Definition or equality by definition
${\left\\{{s_{1},\ldots,s_{n}}\right\\}}$ | Set comprising of elements $s_{1},\ldots,s_{n}$
${\left\lvert{S}\right\rvert}$ or ${\left\lvert{{\left\\{{s_{1},\ldots,s_{n}}\right\\}}}\right\rvert}$ | Cardinality of set $S$
${\operatorname{E}}\left[{X}\right]$ | Mean value of random variable $X$
${\operatorname{Var}}\left[X\right]$ | Variance of random variable $X$
$\gamma_{1}$ | Skewness coefficient

##  2 Previous Work

###

Mobile health applications cover a broad spectrum of cases as listed for
instance in Sunyaev et al. (2014) or in Fox and Duggan (2010). These include
pregnancy as described in Banerjee et al. (2013), heart beat as mentioned in
Steinhubl et al. (2013), and blood pressure as stated in Logan et al. (2007).
Using mobile health applictions results from increased awareness of the
digital health potential as Rich and Miah (2014) claims. A major driver for
the latter is the formation of thematically related communities in online
social media as stated in Ba and Wang (2013). Another factor accounting for
the popularity as well as for the ease of health applications is gamification
as found in Lupton (2013) and Pagoto and Bennett (2013), namely the business
methodologies relying on gaming elements as their names suggest - see for
instance Deterding et al. (2011a), Deterding et al. (2011b), or Huotari and
Hamari (2012). Gamification can already be found at the very core of such
applications as described in Cugelman (2013).

###

The processing path of any digital health may take several forms as shown in
Serbanati et al. (2011). For an overview of recent security practices for
mobile health applications see Papageorgiou et al. (2018). Path analysis as in
Kanavos et al. (2017) play a central role in graph mining in various contexts,
for instance in social networks as in Drakopoulos et al. (2017). Finally, the
advent of advanced GPU technologies can lead to more efficient graph
algorithms as in Drakopoulos et al. (2018).

###

Finally, although it has been only very recently enforced (May 2018), GDPR,
the EU directive governing the collection, processing, and sharing of
sensitive personal information, seems to be already shaping more transparent
conditions the smartphone applications ecosystem is adapting to. In fact,
despite the original protests that GDPR may be excessively constraining under
certain circumstances described in Charitou et al. (2018), consumers seem to
trust mobile applications which clearly outline their intentions concerning
any collected piece of personal information as Bachiri et al. (2018) found
out.

##  3 Architecture

###

This section presents and analyzes the proposed cooperative architecture for
mobile digital health applications. Figure 1 visualizes an instance of a
mobile health application running on a smartphone and a number of peers which
can be reached either by WiFi or by regular mobile services.

Figure 1: Instance of a mobile application surrounded by peers.

###

As with the majority of mobile architectures, the proposed architecture is
conceptually best described with graphs, as concepts such as connectivity and
community structure can be naturally expressed. To this end, the cell phones,
the base stations, and the WiFi access points are represented as vertices,
each device category being represented as a different type. Moreover,
connections between these are represented as edges, where each edge is also of
different type depending on the connection. These can be easily programmed in
a graph database like Neo4j.

###

The general constraints that will be the basis for the subsequent analysis are
as follows:

  * •

Assume that a mobile health application monitoring a biomarker or a biosignal
must deliver results every $T_{0}$ time units, usually seconds. Additionally
assuming that the required computation can be split into $n+1$ parts to be
distributed to the available $n$ neighbors, then:

$T_{a}+T_{p}+T_{s}+2T_{c}\leq T_{0}$ (1)

Where $T_{a}$, $T_{p}$, $T_{s}$, and $T_{c}$ denote respectively the time
required for analysis, namely breaking down the computation and assigning each
neighbor a task, processing, namely the time of the slowest task, synthesis,
namely assemblying the solution of each task to create the general solution,
and communication. The latter term counts twice as the data and the task need
to be communicated and then the results need to be collected.

  * •

In mobile communications is of paramount importance the minimization of the
energy dedicated to a single task. In general the relationship between a given
task and the energy spent for its accomplishment is unknown. However, given
that tasks have a short duration, it is fairly reasonable to assume that the
same function $f{\left({\cdot}\right)}$ links the task and the energy at each
neighbor. Then the following inequality should also be satisfied:

$\displaystyle{\left({n+1}\right)}f{\left({T_{p}}\right)}+f{\left({T_{a}}\right)}+f{\left({T_{s}}\right)}+2{\left({n+1}\right)}f{\left({T_{c}}\right)}$
$\displaystyle\>\leq\>f{\left({T_{0}}\right)}\>\Leftrightarrow\>$ (2)
$\displaystyle\frac{f{\left({T_{0}}\right)}-f{\left({T_{a}}\right)}-f{\left({T_{s}}\right)}}{f{\left({T_{p}}\right)}+2{\left({T_{c}}\right)}}-1$
$\displaystyle\>\geq\>n$ (3)

###

Given the fundamental constraints (1) and (3), let us estimate the key
parameter $T_{c}$, since $T_{a}$, $T_{s}$, and $T_{p}$ depend on the problem
and $T_{0}$ is a constraint.

###

Let $e_{i,j}$ denote the communication link between vertices $v_{i}$ and
$v_{j}$ has a given capacity $C_{i,j}$ as well as a propagation delay
$\tau_{i,j}$. Then, the number of bits $b_{i,j}$ which can be transmitted over
edge $e_{i,j}$ in a time slot of length $\tau_{0}$ is, assuming the variables
are expressed in the proper units:

$b_{i,j}\>=\>C_{i,j}{\left({\tau_{0}-\tau_{i,j}}\right)}$ (4)

If the link delay $\tau_{i,j}$ is expressed as a percentage
$0<\rho^{\tau}_{i,j}<1$ of the time slot $\tau_{0}$, then:

$b_{i,j}\>=\>C_{i,j}\tau_{0}{\left({1-\rho^{\tau}_{i,j}}\right)}$ (5)

Note that the case $\rho^{\tau}_{i,j}=0$ represents a near physical
impossibility, whereas the case $\rho^{\tau}_{i,j}=1$ denotes either a useless
link or a misconfigured network protocol.

###

In a similar way, if $C_{0}$ is the maximum capacity, then each $C_{i,j}$ can
be expressed as a percentage $0<\rho^{C}_{i,j}\leq 1$ of the former. Thus:

$b_{i,j}\>=\>C_{0}\tau_{0}\rho^{C}_{i,j}{\left({1-\rho^{\tau}_{i,j}}\right)}\>=\>B_{0}\rho^{C}_{i,j}{\left({1-\rho^{\tau}_{i,j}}\right)}$
(6)

Note that in this case $\rho^{C}_{i,j}$ can be $1$, unless $C_{0}$ is an
asymptotically upper limit. Therefore, if for the given task $B_{i,j}$ bits
must be transmitted, then the total number of slots for that particular link
is:

$T_{i,j}\>=\>\left\lceil\frac{B_{i,j}}{b_{i,j}}\right\rceil$ (7)

###

At this point, we can estimate $T_{c}$ as:

$T_{c}\>{\buildrel{{\triangle}}\over{=}}\>{\operatorname{E}}\left[{T_{i,j}}\right]$
(8)

###

Furthermore, we can use the distribution of $T_{c}$ to determine whether a big
task should be subdivided to smaller tasks. Assuming $\tau_{0}$ is constant,
then it can be used as a reference point to consider the frequency
distribution of $T_{i,j}$, which can be treated as a probability distribution.

###

For any random variable $X$ is possible to define the skewness coefficient
$\gamma_{1}$ as:

$\gamma_{1}\>{\buildrel{{\triangle}}\over{=}}\>{\operatorname{E}}\left[{\frac{X-{\operatorname{E}}\left[{X}\right]}{\sqrt{{\operatorname{Var}}\left[X\right]}}}\right]\>=\>\frac{{\operatorname{E}}\left[{X^{3}}\right]-3{\operatorname{E}}\left[{X}\right]{\operatorname{Var}}\left[X\right]-{\operatorname{E}}\left[{X}\right]^{3}}{{\operatorname{Var}}\left[X\right]^{\frac{3}{2}}}$
(9)

In equation (9) ${\operatorname{E}}\left[{X}\right]$ and
${\operatorname{Var}}\left[X\right]$ stand for the stochastic mean and
variance of $X$ respectively. In actual settings these can be replaced by
their sample counterparts and they can be updated as new measurements are
collected. In the derivation of the right hand side of (9) the following
properties were used:

$\displaystyle{\operatorname{E}}\left[{\sum_{k=1}^{n}\alpha_{k}X_{k}+\alpha_{0}}\right]$
$\displaystyle\>=\>\sum_{k=1}^{n}\alpha_{k}{\operatorname{E}}\left[{X_{k}}\right]+\alpha_{0}$
(10) $\displaystyle{\operatorname{Var}}\left[\alpha_{1}X+\alpha_{0}\right]$
$\displaystyle\>=\>\alpha_{1}^{2}{\operatorname{Var}}\left[X\right]$ (11)

###

The skewness sign indicates the shape of the distribution. When $\gamma_{1}$
is negative, then $X$ takes larger values with higher probability, whereas
when $\gamma_{1}$ is positive, then $X$ is more likely to take lower values.
Finally, in the case where $\gamma_{1}$ is zero, then the distribution of $X$
is symmetric, as for instance in the case of the binomial distribution.

###

Therefore, positive values of the skewness coefficient $\gamma_{1}$ for the
distribution of $T_{c}$ of the channel delays indicate that it is more likely
more time to be available for useful information transmission.

###

The proposed methodology is summarized in algorithm 1.

0: Knowledge of $T_{0}$, $T_{s}$, $T_{a}$, and $T_{p}$.

0: A cooperative computation takes place.

1: repeat

2: update estimates for ${\left\\{{T_{i,j}}\right\\}}$

3: if equations (3) and (11) are satisfied then

4: break the problem into tasks

5: end if

6: communicate tasks

7: compute tasks

8: collect results

9: compose answer

10: until true

Algorithm 1 The proposed scheme.

##  4 Conclusions

###

This chapter presents a probabilistic architecture for cooperative computation
in mobile health app settings. It relies on a higher order statistical
criterion, namely the skewness coefficient of the number of slots which are
suitable for communication, in order to estimate whether a computation can be
broken into smaller tasks and communicated to neighboring smartphones over
WiFi or the ordinary cell network. Once the tasks are complete, the results
are collected back at the controling smartphone and an answer is generated
using a synthesis of these results.

###

In order to find the hard limits of the proposed architecture and to assess
its performance under various operational scenaria, a number of simulations
must be run in addition to theoretical probabilistic analysis. Additionally,
more conditions should be added to the architecture, for instance what happens
when a neighboring smartphone stops working or is moved out of range.
Moreover, conditions for duplicating certain critical computation must also be
created.

###### Acknowledgements.

This chapter is part of Tensor 451, a long term research initiative whose
primary objective is the development of novel, scalable, numerically stable,
and interpretable tensor analytics.

## References

  * Ba and Wang (2013) Ba S, Wang L (2013) Digital health communities: The effect of their motivation mechanisms. Decision Support Systems 55(4):941–947
  * Bachiri et al. (2018) Bachiri M, Idri A, Fernández-Alemán JL, Toval A (2018) Evaluating the privacy policies of mobile personal health records for pregnancy monitoring. Journal of medical systems 42(8):144
  * Banerjee et al. (2013) Banerjee A, Chen X, Erman J, Gopalakrishnan V, Lee S, Van Der Merwe J (2013) MOCA: A lightweight mobile cloud offloading architecture. In: Proceedings of the eighth ACM international workshop on Mobility in the evolving internet architecture, ACM, pp 11–16
  * Charitou et al. (2018) Charitou C, Kogias DG, Polykalas SE, Patrikakis CZ, Cotoi IC (2018) Use of apps for crime reporting and the EU General Data Protection Regulation. Societal Implications of Community-Oriented Policing and Technology pp 55–61
  * Cugelman (2013) Cugelman B (2013) Gamification: What it is and why it matters to digital health behavior change developers. JMIR Serious Games 1(1)
  * Deterding et al. (2011a) Deterding S, Dixon D, Khaled R, Nacke L (2011a) From game design elements to gamefulness: Defining gamification. In: Proceedings of the 15th international academic MindTrek conference: Envisioning future media environments, ACM, pp 9–15
  * Deterding et al. (2011b) Deterding S, Sicart M, Nacke L, O’Hara K, Dixon D (2011b) Gamification. using game-design elements in non-gaming contexts. In: CHI’11 extended abstracts on human factors in computing systems, ACM, pp 2425–2428
  * Drakopoulos et al. (2017) Drakopoulos G, Kanavos A, Mylonas P, Sioutas S (2017) Defining and evaluating Twitter influence metrics: A higher order approach in Neo4j. SNAM 71(1)
  * Drakopoulos et al. (2018) Drakopoulos G, Liapakis X, Tzimas G, Mylonas P (2018) A graph resilience metric based on paths: Higher order analytics with GPU. In: ICTAI, IEEE
  * Fox and Duggan (2010) Fox S, Duggan M (2010) Mobile health 2010. Pew Internet and American Life Project Washington, DC
  * Huotari and Hamari (2012) Huotari K, Hamari J (2012) Defining gamification: a service marketing perspective. In: Proceedings of the 16th international academic MindTrek conference, ACM, pp 17–22
  * Kanavos et al. (2017) Kanavos A, Drakopoulos G, Tsakalidis A (2017) Graph community discovery algorithms in Neo4j with a regularization-based evaluation metric. In: WEBIST
  * Logan et al. (2007) Logan AG, et al. (2007) Mobile phone–based remote patient monitoring system for management of hypertension in diabetic patients. American Journal of Hypertension 20(9):942–948
  * Lupton (2013) Lupton D (2013) The digitally engaged patient: Self-monitoring and self-care in the digital health era. Social Theory and Health 11(3):256–270
  * Pagoto and Bennett (2013) Pagoto S, Bennett GG (2013) How behavioral science can advance digital health. Translational behavioral medicine 3(3):271–276
  * Papageorgiou et al. (2018) Papageorgiou A, Strigkos M, Politou E, Alepis E, Solanas A, Patsakis C (2018) Security and privacy analysis of mobile health applications: The alarming state of practice. IEEE Access 6:9390–9403
  * Rich and Miah (2014) Rich E, Miah A (2014) Understanding digital health as public pedagogy: A critical framework. Societies 4(2):296–315
  * Serbanati et al. (2011) Serbanati LD, Ricci FL, Mercurio G, Vasilateanu A (2011) Steps towards a digital health ecosystem. Journal of Biomedical Informatics 44(4):621–636
  * Steinhubl et al. (2013) Steinhubl SR, Muse ED, Topol EJ (2013) Can mobile health technologies transform health care? JAMA 310(22):2395–2396
  * Sunyaev et al. (2014) Sunyaev A, Dehling T, Taylor PL, Mandl KD (2014) Availability and quality of mobile health app privacy policies. Journal of the American Medical Informatics Association 22(e1):e28–e33


# A four-mean theorem and its application to pseudospectra

Thomas Ransford  Département de mathématiques et statistique, Université
Laval, Québec (QC) G1V 0A6, Canada<EMAIL_ADDRESS>and  Nathan Walsh
Département de mathématiques et statistique, Université Laval, Québec (QC) G1V
0A6, Canada<EMAIL_ADDRESS>
(Date: 9 July 2022)

###### Abstract.

Let $N$ be an integer with $N\geq 4$. We show that, if $x_{1},\dots,x_{N}$ and
$y_{1},\dots,y_{N}$ are $N$-tuples of strictly positive numbers whose
arithmetic, geometric and harmonic means agree, then

$\max_{j}x_{j}<(N-2)\max_{j}y_{j}\quad\text{and}\quad\min_{j}x_{j}<(N-2)\min_{j}y_{j}.$

A generalized version of this result (where some of $x_{j}$ and $y_{j}$ are
allowed to be zero) is used to show that, if $N\geq 4$ and $A,B$ are $N\times
N$ matrices with super-identical pseudospectra, then, for every polynomial
$p$, we have

$\|p(A)\|<\sqrt{N-2}\|p(B)\|,$

unless $p(A)=p(B)=0$. This improves a previously known inequality to the point
of being sharp, at least for $N=4$.

###### Key words and phrases:

Arithmetic mean, Geometric mean, Harmonic mean, Pseudospectrum, Singular value

###### Key words and phrases:

Arithmetic mean, Geometric mean, Harmonic mean, Pseudospectrum, Singular value

###### 2010 Mathematics Subject Classification:

Primary 26E20; Secondary 15A18, 26D15

###### 2010 Mathematics Subject Classification:

Primary 26E20; Secondary 15A18, 26D15

Ransford supported by grants from NSERC and the Canada Research Chairs
program. Walsh supported by an NSERC Undergraduate Student Research Award and
an FRQNT Supplement.

##  1\. Introduction and statement of results

Our main result is the following theorem.

######  Theorem 1.1.

Let $N\geq 4$ and let $x_{1},\dots,x_{N},y_{1},\dots,y_{N}\geq 0$. If

(1) $\displaystyle\sum_{j=1}^{N}x_{j}$ $\displaystyle=\sum_{j=1}^{N}y_{j},$
(2) $\displaystyle\prod_{j=1}^{N}x_{j}$ $\displaystyle=\prod_{j=1}^{N}y_{j},$
and (3) $\displaystyle\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}x_{j}$
$\displaystyle=\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}y_{j},$

then

(4) $\max_{j}x_{j}\leq(N-2)\max_{j}y_{j}.$

Equality holds in (4) if and only if there exists $c\geq 0$ such that, after
rearranging the $x_{j}$ and $y_{j}$ in decreasing order, we have

$(x_{1},\dots,x_{N})=c(N-2,0,\dots,0)\quad\text{and}\quad(y_{1},\dots,y_{N})=c(1,\dots,1,0,0).$

######  Corollary 1.2.

Let $N\geq 4$. If $x_{1},\dots,x_{N}$ and $y_{1},\dots,y_{N}$ are $N$-tuples
of strictly positive numbers whose arithmetic, geometric and harmonic means
agree, then

(5)
$\max_{j}x_{j}<(N-2)\max_{j}y_{j}\quad\text{and}\quad\min_{j}x_{j}<(N-2)\min_{j}y_{j}.$

###### Proof.

If all the $x_{j}$ and $y_{j}$ are strictly positive, then, under condition
(2), the condition (3) is equivalent to the statement that
$\sum_{j=1}^{N}1/x_{j}=\sum_{j=1}^{N}1/y_{j}$, in other words, that the
harmonic means agree. Therefore the inequality between maxima in (5) is a
consequence of Theorem 1.1.

The inequality between minima follows by applying the one for maxima to the
$N$-tuples $(1/y_{1},\dots,1/y_{N})$ and $(1/x_{1},\dots,1/x_{N})$, whose
arithmetic, geometric and harmonic means also agree. ∎

######  Remarks.

(i) Corollary 1.2 shows that, if the $x_{j}$ and $y_{j}$ are constrained so
that three of their means (arithmetic, geometric, harmonic) agree, then the
fourth (maximum or minimum) satisfies the inequality (5). For this reason, we
have dubbed it (and by extension Theorem 1.1) a ‘four-mean theorem’.

(ii) The more complicated formulation of the condition (3) allows for the
possibility that some of the $x_{j}$ and $y_{j}$ are zero. This is useful
since it contains the case of equality.

(iii) Theorem 1.1 is only interesting if $N\geq 4$ since, if $N=3$, then the
conditions (1), (2) and (3) already imply that $x_{j}$ are a permutation of
the $y_{j}$. Indeed, they tell us that the polynomials
$(t-x_{1})(t-x_{2})(t-x_{3})$ and $(t-y_{1})(t-y_{2})(t-y_{3})$ have the same
coefficients, and therefore the same roots.

(iv) There are also two-mean and three-mean theorems. In fact it is obvious
that if just (1) holds, then (4) is true with $(N-2)$ replaced by $N$, and
this is optimal. Also, if (1) and (2) both hold, then (4) is true with $(N-2)$
replaced by $(N-1)$, and again this is optimal. However, this is less obvious,
and indeed this three-mean theorem is a step on the route to establishing
Theorem 1.1.

Our motivation for studying constraints and inequalities such as those in
Theorem 1.1 arises from a problem in the theory of pseudospectra of matrices,
which we now describe. For background on pseudospectra, we refer to the book
[4] and the survey article [3].

It is known that the pseudospectra of a matrix $A$, namely the level sets of
$\|(A-zI)^{-1}\|$, do not suffice to determine the operator norm of
polynomials of $A$, see e.g. [4, Section 47] and [3, Section 2], in particular
[3, Theorem 2.3]. In an attempt to overcome this problem, the authors of [2]
proposed looking at not only the lowest singular values of $A-zI$ for
$z\in\mathbb{C}$ (namely the reciprocals of $\|(A-zI)^{-1}\|$), but all the
singular values. They showed that this information is sufficient to determine
the operator norm of any polynomial of $A$, up to a factor depending only on
the dimension.

Here is a more precise formulation of their result. We say that two complex
$N\times N$ matrices $A,B$ have _super-identical pseudospectra_ if

$s_{j}(A-zI)=s_{j}(B-zI)\quad(j=1,2,\dots,N,~{}z\in\mathbb{C}),$

where $s_{1},\dots,s_{N}$ denote the singular values, ordered so that
$s_{1}\geq\cdots\geq s_{N}$. It was shown in [2, Theorem 1.3] that, if $A,B$
are $N\times N$ matrices with super-identical pseudospectra and $p$ is a
polynomial, then

(6) $\|p(A)\|\leq\sqrt{N}\|p(B)\|.$

It is not obvious, _a priori_ , whether the constant $\sqrt{N}$ can be
replaced by $1$ in this inequality. In fact it can if $N=1,2,3$ (see [3,
Section 4]), but an example constructed in [2] shows that this is no longer
the case when $N\geq 4$. However, the authors of [2] mentioned that they did
not know whether the bound $\sqrt{N}$ in (6) is optimal. The following result
shows that it is not, and replaces it with a bound that is sharp, at least
when $N=4$.

######  Theorem 1.3.

Let $N\geq 4$ and let $A,B$ be complex $N\times N$ matrices with super-
identical pseudospectra. Then, for any polynomial $p$, we have

(7) $\|p(A)\|<\sqrt{N-2}\|p(B)\|,$

unless $p(A)=p(B)=0$. The constant $\sqrt{N-2}$ is sharp at least if $N=4$.

The proof of (7) is based on the four-mean theorem, Theorem 1.1. The proof of
sharpness in the case $N=4$ is based on the example from [2] mentioned above,
which will be described in detail in Section 5 below. The same example also
permits us to deduce a related result, which we now describe.

Armentia, Gracia and Velasco [1] showed that, if $A,B$ have super-identical
pseudospectra, then they are similar, in other words, there exists an
invertible matrix $W$ such that

(8) $B=W^{-1}AW.$

In this case, $p(B)=W^{-1}p(A)W$ for every polynomial $p$, and so

$\|p(B)\|\leq\|W^{-1}\|\|p(A)\|\|W\|.$

It is thus tempting to believe that (7) and (8) can be subsumed in a single
result in which (8) holds with $\|W\|\|W^{-1}\|\leq\sqrt{N-2}$. Even if this
is false, one might hope that, at the very least, $W$ may be chosen so that
$\|W\|\|W^{-1}\|\leq C(N)$ for some constant $C(N)$ depending only on $N$. The
following theorem shows that, perhaps surprisingly, there is no such result.

######  Theorem 1.4.

Given $M>0$, there exist $4\times 4$ matrices $A,B$ with super-identical
pseudospectra such that

(9) $\inf\Bigl{\\{}\|W\|\|W^{-1}\|:B=W^{-1}AW\Bigr{\\}}>M.$

The rest of the paper is organized as follows. In Section 2 we establish the
three-mean theorem mentioned earlier, which is then used in Section 3 to prove
the four-mean theorem, Theorem 1.1. Theorem 1.3 is deduced in Section 4,
except for the sharpness statement, which is established in Section 5, where
Theorem 1.4 is also proved.

##  2\. Three-mean theorem

Our goal in this section is to establish the following theorem.

######  Theorem 2.1.

Let $N\geq 3$ and let $x_{1},\dots,x_{N},y_{1},\dots,y_{N}\geq 0$. If

$\sum_{j=1}^{N}x_{j}=\sum_{j=1}^{N}y_{j}\quad\text{and}\quad\prod_{j=1}^{N}x_{j}=\prod_{j=1}^{N}y_{j},$

then

(10) $\max_{j}x_{j}\leq(N-1)\max_{j}y_{j}.$

Equality holds in (10) if and only if there exists $c\geq 0$ such that, after
rearranging the $x_{j}$ and $y_{j}$ in decreasing order, we have

$(x_{1},\dots,x_{N})=c(N-1,0,\dots,0)\quad\text{and}\quad(y_{1},\dots,y_{N})=c(1,\dots,1,1,0).$

We shall prove this theorem by reformulating it as an optimization result.
Since the result obviously holds if all the numbers $y_{j}$ are equal to zero,
we can suppose that at least one of them is non-zero. Normalizing so that
$\max_{j}x_{j}=x_{1}$ and $\max_{j}y_{j}=y_{1}=1$, we are led to consider the
following problem.

######  Problem 2.2.

Let $N\geq 3$. Maximize $x_{1}$ subject to the following constraints:

(11) $\left\\{\begin{aligned} &x_{j}\geq 0~{}(j=1,\dots,N),\\\
&y_{1}=1,~{}0\leq y_{j}\leq 1~{}(2\leq j\leq N),\\\
&\sum_{j=1}^{N}x_{j}=\sum_{j=1}^{N}y_{j},\\\
&\prod_{j=1}^{N}x_{j}=\prod_{j=1}^{N}y_{j}.\end{aligned}\right.$

Here is the solution to Problem 2.2, which establishes Theorem 2.1.

######  Theorem 2.3.

Let $N\geq 3$. The maximum value of $x_{1}$ subject to the constraints (11) is
$N-1$, attained uniquely when $x_{2}=\cdots=x_{N}=0$ and all but one of the
$y_{j}$ are equal to $1$, the remaining one being equal to $0$.

###### Proof.

Let $S$ be the set of
$(\mathbf{x},\mathbf{y})\in\mathbb{R}^{N}\times\mathbb{R}^{N}$ that satisfy
the constraints (11). Clearly the numbers $y_{j}$ lie in $[0,1]$. Also it is
easy to see that the numbers $x_{j}$ lie in $[0,N]$. Therefore $S$ is a
compact set. The function $(\mathbf{x},\mathbf{y})\mapsto x_{1}$ is
continuous, so it attains its maximum on $S$, say at
$(\mathbf{x}^{*},\mathbf{y}^{*})$.

If $\mathbf{x}=(N-1,0,\dots,0)$ and $\mathbf{y}=(1,\dots,1,0)$, then
$(\mathbf{x},\mathbf{y})\in S$ and $x_{1}=N-1$. Thus we certainly have
$x_{1}^{*}\geq N-1$.

We shall show by contradiction that at least one of the terms $x_{j}^{*}$ or
$y_{j}^{*}$ is equal to zero. Suppose, if possible, that $x_{j}^{*}>0$ and
$y_{j}^{*}>0$ for all $j$. Applying the standard Lagrange-multiplier argument
to

$x_{1}+\alpha\Bigl{(}\sum_{j=1}^{N}x_{j}-\sum_{j=1}^{N}y_{j}\Bigr{)}+\beta\Bigl{(}\sum_{j=1}^{N}\log
x_{j}-\sum_{j=1}^{N}\log y_{j}\Bigr{)},$

we see that

$\left\\{\begin{aligned} 1+\alpha+\beta/x^{*}_{1}&=0,\\\
\alpha+\beta/x^{*}_{j}&=0,\quad(2\leq j\leq N),\\\
\alpha+\beta/y^{*}_{j}&=0,\quad(1\leq j\leq
N,~{}\text{if~{}}y^{*}_{j}<1).\end{aligned}\right.$

From the first equation, $\alpha$ and $\beta$ cannot both be zero, and from
the second, they are in fact both non-zero. Writing $u:=-\beta/\alpha$, we
deduce that

$\left\\{\begin{aligned} x_{j}^{*}&=u\quad(2\leq j\leq N),\\\
y_{j}^{*}&=u\text{~{}or~{}}1\quad(1\leq j\leq N).\end{aligned}\right.$

There are now two possibilities. The first is that all the $y_{j}^{*}$ are
equal to $1$. In this case, the arithmetic and geometric means of the
$y^{*}_{j}$ are equal to $1$, so by (11) the arithmetic and geometric means of
the $x^{*}_{j}$ are also equal to $1$. By the case of equality in the AM-GM
inequality, this forces all the $x^{*}_{j}$ to be equal to $1$. This
contradicts the fact that $x_{1}^{*}\geq N-1$.

The second possibility is that some $y_{j}^{*}=u<1$. This implies that

$N-1+(N-1)u\leq\sum_{j=1}^{N}x_{j}^{*}=\sum_{j=1}^{N}y_{j}^{*}\leq N-1+u.$

Since $N\geq 3$ and $u>0$, this is impossible.

Thus both possibilities lead to contradictions. We conclude that at least one
of the terms $x_{j}^{*}$ or $y_{j}^{*}$ is equal to zero, as claimed.

Since $\prod_{j=1}^{N}x_{j}^{*}=\prod_{j=1}^{N}y_{j}^{*}=0$, at least one
$y_{j}^{*}=0$, so $\sum_{j=1}^{N}y^{*}_{j}\leq N-1$. It follows that

$N-1\leq x_{1}^{*}\leq\sum_{j=1}^{N}x^{*}_{j}=\sum_{j=1}^{N}y^{*}_{j}\leq
N-1.$

Therefore we have equality throughout, which shows that $x_{1}^{*}=N-1$ and
$x_{j}^{*}=0$ for all $j\geq 2$, and also that all but one of the $y_{j}^{*}$
satisfy $y_{j}^{*}=1$. This concludes the proof. ∎

##  3\. Four-mean theorem

Following the idea of the preceding section, we shall prove Theorem 1.1 by
formulating it as the solution to an optimization problem. Here is the
problem:

######  Problem 3.1.

Let $N\geq 4$. Maximize $x_{1}$ subject to the following constraints:

(12) $\left\\{\begin{aligned} &x_{j}\geq 0~{}(j=1,\dots,N),\\\
&y_{1}=1,~{}0\leq y_{j}\leq 1~{}(2\leq j\leq N),\\\
&\sum_{j=1}^{N}x_{j}=\sum_{j=1}^{N}y_{j},\\\
&\prod_{j=1}^{N}x_{j}=\prod_{j=1}^{N}y_{j},\\\
&\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}x_{j}=\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}y_{j}.\end{aligned}\right.$

And here is the solution.

######  Theorem 3.2.

Let $N\geq 4$. The maximum value of $x_{1}$ subject to the constraints (12) is
$N-2$, attained uniquely when $x_{2}=\cdots=x_{N}=0$ and all but two of the
$y_{j}$ are equal to $1$, the remaining ones being equal to $0$.

Before embarking upon the main proof, it will be convenient to separate out
some algebraic results needed in the course of the argument.

######  Lemma 3.3.

Let $N\geq 4$ and let $r\in\\{1,2,\dots,N-1\\}$.

  1. (i)

If

$f(t):=t^{2N-2}-(N-1)t^{N}+(N-1)t^{N-2}-1,$

then $f(t)\neq 0$ for all $t>0$ except $t=1$.

  2. (ii)

If

$g(u,v):=(N-1)u^{N}-(N-r)u^{N-1}-rvu^{N-1}+v^{r},$

then $g(u,v)\neq 0$ in $0<u<v<1$.

###### Proof.

(i) A direct calculation shows that
$f(1)=f^{\prime}(1)=f^{\prime\prime}(1)=0$, in other words, that $f$ has a
triple zero at $t=1$. By Descartes’ rule of signs, $f$ has at most three zeros
in $(0,\infty)$, counted according to multiplicity. It follows that $f$ has no
zeros in $(0,\infty)$ other than $1$.

(ii) On the diagonal $u=v$, we have

$g(u,u)=(N-1-r)u^{N}-(N-r)u^{N-1}+u^{r}.$

This is identically zero if $r=N-1$. Suppose that $r<N-1$. Then, by Descartes’
rule of signs again, the right-hand side has at most two zeros in
$(0,\infty)$. On the other hand, a direct verification shows that the right-
hand side has a double zero at $u=1$. Therefore $g(u,u)\neq 0$ for all
$u\in(0,1)$. Since $g(u,u)=u^{r}(1+o(u))$ as $u\to 0$, it follows that
$g(u,u)>0$ for all $u\in(0,1)$. Putting together the cases $r=N-1$ and
$r<N-1$, we obtain

(13) $g(u,u)\geq 0\quad(0<u<1).$

Now, a simple computation gives

(14) $\frac{\partial g}{\partial v}(u,v)=-ru^{N-1}+rv^{r-1}>0\quad(0<u<v<1).$

Finally, (13) and (14) together imply that $g(u,v)>0$ in $0<u<v<1$. ∎

###### Proof of Theorem 3.2.

The proof follows the same general lines as that of Theorem 2.1, though the
details are a bit more involved.

Let $S$ be the set of
$(\mathbf{x},\mathbf{y})\in\mathbb{R}^{N}\times\mathbb{R}^{N}$ obeying the
constraints (12). Then $S$ is a compact set, so the function
$(\mathbf{x},\mathbf{y})\mapsto x_{1}$ attains its maximum on $S$, say at
$(\mathbf{x}^{*},\mathbf{y}^{*})$.

If $\mathbf{x}=(N-2,0,\dots,0)$ and $\mathbf{y}=(1,\dots,1,0,0)$, then
$(\mathbf{x},\mathbf{y})\in S$ and $x_{1}=N-2$, so we certainly have
$x_{1}^{*}\geq N-2$.

We claim that at least one of the $x_{j}^{*}$ or $y_{j}^{*}$ is equal to zero.
To prove the claim, we proceed by contradiction. So, let us suppose, if
possible, that $x_{j}^{*}>0$ and $y_{j}^{*}>0$ for all $j$.

In this situation, the final constraint in (12) is equivalent to the condition
that $\sum_{j=1}^{N}1/x_{j}^{*}=\sum_{j=1}^{N}1/y_{j}^{*}$. We may therefore
apply the standard Lagrange-multiplier argument to

$x_{1}+\alpha\Bigl{(}\sum_{j=1}^{N}x_{j}-\sum_{j=1}^{N}y_{j}\Bigr{)}+\beta\Bigl{(}\sum_{j=1}^{N}\log
x_{j}-\sum_{j=1}^{N}\log
y_{j}\Bigr{)}+\gamma\Bigl{(}\sum_{j=1}^{N}\frac{1}{x_{j}}-\sum_{j=1}^{N}\frac{1}{y_{j}}\Bigr{)}$

to obtain

$\left\\{\begin{aligned}
1+\alpha+\beta/x^{*}_{1}-\gamma/(x_{1}^{*})^{2}&=0,\\\
\alpha+\beta/x^{*}_{j}-\gamma/(x_{j}^{*})^{2}&=0,\quad(2\leq j\leq N),\\\
\alpha+\beta/y^{*}_{j}-\gamma/(y_{j}^{*})^{2}&=0,\quad(1\leq j\leq
N,~{}\text{if~{}}y^{*}_{j}<1).\end{aligned}\right.$

The first equality shows that the constants $\alpha,\beta,\gamma$ are not all
zero, and the remaining equalities show that

$\left\\{\begin{aligned} x_{j}^{*}&=u\text{~{}or~{}}v\quad(2\leq j\leq N),\\\
y_{j}^{*}&=u\text{~{}or~{}}v\text{~{}or~{}}1\quad(1\leq j\leq
N),\end{aligned}\right.$

where $u$ and $v$ are the roots of $\alpha t^{2}+\beta t-\gamma=0$.

If some $x_{j}^{*}$ is equal to some $y_{k}^{*}$, then the vectors
$\mathbf{x}^{\prime},\mathbf{y}^{\prime}\in\mathbb{R}^{N-1}$ formed by the
remaining components satisfy the hypotheses of Theorem 2.1 (with $N$ replaced
by $N-1$). By that theorem, we deduce that $x_{1}^{\prime}\leq N-2$. Since
$x_{1}^{\prime}=x_{1}^{*}\geq N-2$, we actually have equality throughout. By
the case of equality in Theorem 2.1, all the remaining $x_{j}^{\prime}=0$ ,
which contradicts the supposition that $x_{j}^{*}>0$ for all $j$. We are thus
led to conclude that, in fact,

$\left\\{\begin{aligned} x_{j}^{*}&=u\quad(2\leq j\leq N),\\\
y_{j}^{*}&=v\text{~{}or~{}}1\quad(1\leq j\leq N).\end{aligned}\right.$

If all the numbers $y_{j}^{*}$ are equal to $1$, then the arithmetic and
geometric means of the $y^{*}_{j}$ are equal to $1$, so by (12) the arithmetic
and geometric means of the $x^{*}_{j}$ are also equal to $1$, which forces all
the $x^{*}_{j}$ to be equal to $1$. This contradicts the fact that
$x_{1}^{*}\geq N-2$. We conclude that there exists an integer $r$ with $1\leq
r\leq N-1$ such that exactly $r$ of the $y_{j}^{*}$ are equal to $v$ and the
remaining $(N-r)$ are equal to $1$. The constraints (12) then become

(15) $\displaystyle x_{1}^{*}+(N-1)u$ $\displaystyle=N-r+rv,$ (16)
$\displaystyle x_{1}^{*}u^{N-1}$ $\displaystyle=v^{r},$ (17)
$\displaystyle\frac{1}{x_{1}^{*}}+\frac{N-1}{u}$
$\displaystyle=N-r+\frac{r}{v}.$

We also have $u\neq v$ and $v<1$.

The argument now subdivides into two cases, according to whether $u<v$ or
$u>v$.

_Case I:_ $u<v$. Eliminating $x_{1}^{*}$ from (15) and (16) gives

(18) $v^{r}+(N-1)u^{N}=(N-r)u^{N-1}+rvu^{N-1}.$

In other words, $g(u,v)=0$, where $g$ is the function in Lemma 3.3 (ii). But
by that lemma, $g(u,v)\neq 0$ for $0<u<v<1$. This contradiction concludes the
argument for Case I.

_Case II:_ $u>v$. In this case we must have $r=1$. Indeed, by (15), we have

$N-r+rv=x_{1}^{*}+(N-1)u\geq N-2+(N-1)v,$

which, after simplification, leads to

$(N-1-r)v\leq 2-r.$

Since the left-hand side is non-negative, we must have $r\leq 2$. Also, if
$r=2$, then the left-hand side is zero, which implies that $r=N-1$,
contradicting the fact that $N\geq 4$. The only remaining possibility is that
$r=1$, as claimed.

Multiplying together (15) and (17), and recalling that $r=1$, we obtain

$1+(N-1)^{2}+(N-1)\frac{x_{1}^{*}}{u}+(N-1)\frac{u}{x_{1}^{*}}=(N-1)^{2}+1+\frac{N-1}{v}+(N-1)v,$

which, after simplification, becomes

$\frac{x_{1}^{*}}{u}+\frac{u}{x_{1}^{*}}=v+\frac{1}{v}.$

Since the function $t\mapsto(t+1/t)$ is $2$-to-$1$ on $t>0$, it follows that
either $x_{1}^{*}/u=v$ or $x_{1}^{*}/u=1/v$. In the first case, (16) implies
that $vu^{N}=v$, which in turn implies that $u=1$ and $x_{1}^{*}=v<1$, a
contradiction. So we must have $x_{1}^{*}=u/v$. Substituting this information
into (16), we find that $u^{N}=v^{2}$ and $x_{1}^{*}=u^{1-N/2}$. Substituting
this into (15), and rearranging, we obtain

$u^{N-1}-(N-1)u^{N/2}+(N-1)u^{N/2-1}-1=0,$

in other words, $f(u^{1/2})=0$, where $f$ is the polynomial in Lemma 3.3 (i).
By that lemma, $f(t)\neq 0$ for all $t>0$ except $t=1$. We conclude that
$u=1$, and hence that $x_{1}^{*}=1$, contradicting the fact that
$x_{1}^{*}\geq N-2$. This concludes the argument for Case II.

Thus, whichever case we are in, we arrive at a contradiction. This shows that,
as claimed, at least one of the $x_{j}^{*}$ or $y_{j}^{*}$ is equal to zero.

Because their geometric means are equal, both vectors
$\mathbf{x}^{*},\mathbf{y}^{*}$ contain a component equal to zero. The vectors
$\mathbf{x}^{\prime},\mathbf{y}^{\prime}\in\mathbb{R}^{N-1}$ formed by the
remaining components then satisfy the hypotheses of Theorem 2.1 (with $N$
replaced by $N-1$). By that theorem, we deduce that $x_{1}^{\prime}\leq N-2$.
Since $x_{1}^{\prime}=x_{1}^{*}\geq N-2$, we actually have equality
throughout. By the case of equality in Theorem 2.1, all the remaining
$x_{j}^{\prime}$ are equal to $0$ and all but one of the $y_{j}^{\prime}$ are
equal to $1$, the remaining one being equal to zero. We conclude that
$x_{1}^{*}=N-2$, all the remaining $x_{j}^{*}$ are equal to $0$ and all but
two of the $y_{j}^{*}$ are equal to $1$, the remaining ones being equal to
zero. This completes the proof of Theorem 3.2, and with it, that of Theorem
1.1. ∎

##  4\. Super-identical pseudospectra

The proof of Theorem 1.3 is based on Theorem 1.1 and the following lemma. As
before, we write $s_{1},\dots,s_{N}$ to denote the singular values of an
$N\times N$ matrix, ordered so that $s_{1}\geq\dots\geq s_{N}$.

######  Lemma 4.1.

Let $N\geq 4$ and let $A,B$ be complex $N\times N$ matrices with super-
identical pseudospectra. Then, for every polynomial $p$,

(19) $\displaystyle\sum_{j=1}^{N}s_{j}(p(A))^{2}$
$\displaystyle=\sum_{j=1}^{N}s_{j}(p(B))^{2},$ (20)
$\displaystyle\prod_{j=1}^{N}s_{j}(p(A))^{2}$
$\displaystyle=\prod_{j=1}^{N}s_{j}(p(B))^{2},$ and (21)
$\displaystyle\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}s_{j}(p(A))^{2}$
$\displaystyle=\sum_{k=1}^{N}\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{N}s_{j}(p(B))^{2}.$

###### Proof.

The equality (19) was already obtained in the course of the proof of (6), see
[2, pp. 516–517]. We do not repeat the argument here.

The equality (20) is a consequence of the result of Armentia, Gracia and
Velasco mentioned earlier, according to which matrices with super-identical
pseudospectra are always similar. Thus $A$ and $B$ are similar, hence also
$p(A)$ and $p(B)$. In particular, $\det(p(A))=\det(p(B))$. Since the absolute
value of the determinant is the product of the singular values, we deduce that
(20) holds.

To establish (21), let us first consider the case when $p(A)$ and $p(B)$ are
invertible. Then $p$ has no common zeros with the characteristic polynomial
$f$ of $A$, so there exist polynomials $q,r$ such that $pq+fr=1$. Since
$f(A)=0$, it follows that $p(A)q(A)=I$. As $A,B$ are similar, $f(B)=0$ as
well, and so also $p(B)q(B)=I$. By (19), with $p$ replaced by $q$, we have

$\sum_{j=1}^{N}s_{j}(q(A))^{2}=\sum_{j=1}^{N}s_{j}(q(B))^{2}.$

But also we have $q(A)=p(A)^{-1}$, and the singular values of $p(A)^{-1}$ are
$1/s_{N}(p(A)),\dots,1/s_{1}(p(A))$. Likewise for $B$. It follows that

$\sum_{j=1}^{N}\frac{1}{s_{j}(p(A))^{2}}=\sum_{j=1}^{N}\frac{1}{s_{j}(p(B))^{2}}.$

Multiplying this equation by equation (20), we obtain (21). This proves (21)
in the case when $p(A)$ and $p(B)$ are invertible. The general case follows by
replacing $p(z)$ by $p(z)+\epsilon$, where $\epsilon\neq 0$, and then letting
$\epsilon\to 0$. ∎

###### Proof of Theorem 1.3.

Let $A,B$ have super-identical pseudospectra, and let $p$ be a polynomial.
Then Lemma 4.1 implies that the non-negative numbers $x_{j}:=s_{j}(p(A))^{2}$
and $y_{j}:=s_{j}(p(B))^{2}$ satisfy the relations (1), (2) and (3) of Theorem
1.1. By that theorem, it follows that

$\max_{j}s_{j}(p(A))^{2}\leq(N-2)\max_{j}s_{j}(p(B))^{2},$

in other words, that the operator norms of $p(A)$ and $p(B)$ satisfy

(22) $\|p(A)\|^{2}\leq(N-2)\|p(B)\|^{2}.$

If we have equality in (22), then, by the case of equality in Theorem 1.1,
there exists $c\geq 0$ such that

$\displaystyle(s_{1}(A)^{2},\dots,s_{N}(A)^{2})$
$\displaystyle=c(N-2,0,\dots,0),$
$\displaystyle(s_{1}(B)^{2},\dots,s_{N}(B)^{2})$
$\displaystyle=c(1,\dots,1,0,0).$

However, since $A,B$ are similar, $p(A)$ and $p(B)$ have the same rank, so
they have exactly the same number of non-zero singular values. This can only
happen if $c=0$. Thus equality holds in (22) if and only if $p(A)=p(B)=0$.

This completes the proof of Theorem 1.3 except for the sharpness statement,
which will be treated in the next section. ∎

##  5\. Sharpness results

Both the sharpness statement in Theorem 1.3 and the negative result about
quantitative similarity, Theorem 1.4, are consequences of the example
contained in the following proposition.

######  Proposition 5.1.

Let $\alpha,\beta\in(0,\pi/4]$, and let

$A=\begin{pmatrix}0&\sec\alpha&0&1\\\ 0&0&\sec\beta\csc\beta&0\\\
0&0&0&\csc\alpha\\\ 0&0&0&0\\\
\end{pmatrix},~{}B=\begin{pmatrix}0&\sec\beta&0&1\\\
0&0&\sec\alpha\csc\alpha&0\\\ 0&0&0&\csc\beta\\\ 0&0&0&0\\\ \end{pmatrix}.$

Then $A,B$ have super-identical pseudospectra, and

(23)
$\frac{s_{1}(A^{2})}{s_{1}(B^{2})}=\frac{\cos\alpha}{\cos\beta}\quad\text{and}\quad\frac{s_{2}(A^{2})}{s_{2}(B^{2})}=\frac{\sin\alpha}{\sin\beta}.$

###### Proof.

The matrices $A,B$ are taken from [2, Theorem 5.1], where it is shown that
they have super-identical pseudospectra. It remains to establish (23).

A calculation gives

$A^{2}=\begin{pmatrix}0&0&\sec\alpha\sec\beta\csc\beta&0\\\
0&0&0&\csc\alpha\sec\beta\csc\beta\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{pmatrix}.$

Since $\alpha\in(0,\pi/4]$, we have $\sec\alpha\leq\csc\alpha$, whence

$s_{1}(A^{2})=\csc\alpha\sec\beta\csc\beta\quad\text{and}\quad
s_{2}(A^{2})=\sec\alpha\sec\beta\csc\beta.$

Similarly

$s_{1}(B^{2})=\csc\beta\sec\alpha\csc\alpha\quad\text{and}\quad
s_{2}(B^{2})=\sec\beta\sec\alpha\csc\alpha.$

The result follows. ∎

###### Proof of the sharpness statement in Theorem 1.3.

Taking $\beta=\pi/4$ and $\alpha$ close to $0$ in Proposition 5.1, we see
that, given $\epsilon>0$, there exist $4\times 4$ matrices $A,B$ with super-
identical pseudospectra such that $\|A^{2}\|/\|B^{2}\|>\sqrt{2}-\epsilon$.
This demonstrates that, if $N=4$, then the constant $\sqrt{N-2}$ in (7) is
sharp. ∎

###### Proof of Theorem 1.4.

Given $M>0$, choose $\beta:=\pi/4$ and $\alpha>0$ sufficiently small so that
$\sin\beta/\sin\alpha>M$. Let $A,B$ be the $4\times 4$ matrices with super-
identical pseudospectra furnished by Proposition 5.1.

If $W$ is an invertible $N\times N$ matrix such that $B=W^{-1}AW$, then
$p(B)=W^{-1}p(A)W$ for every polynomial $p$. It follows that

$s_{j}(p(B))\leq\|W^{-1}\|s_{j}(p(A))\|W\|$

for all polynomials $p$ and all $j\in\\{1,2,3,4\\}$. In particular, we have

$\|W\|\|W^{-1}\|\geq\frac{s_{2}(B^{2})}{s_{2}(A^{2})}=\frac{\sin\beta}{\sin\alpha}>M.$

This proves (9) and establishes the result. ∎

## References

  * [1] G. Armentia, J.-M. Gracia, and F. E. Velasco, _Identical pseudospectra of any geometric multiplicity_ , Linear Algebra Appl. 436 (2012), no. 6, 1683–1688.
  * [2] M. Fortier Bourque and T. Ransford, _Super-identical pseudospectra_ , J. Lond. Math. Soc. (2) 79 (2009), no. 2, 511–528.
  * [3] T. Ransford, _Pseudospectra and matrix behaviour_ , Banach algebras 2009, Banach Center Publ., vol. 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 327–338.
  * [4] L. N. Trefethen and M. Embree, _Spectra and pseudospectra_ , Princeton University Press, Princeton, NJ, 2005.