# LLMs for Test Input Generation for Semantic Caches

Zafaryab Rasool, Scott Barnett, David Willie, Stefanus Kurniawan, Sherwin
Balugo, Srikanth Thudumu, Mohamed Abdelrazek  zafaryab.rasool, scott.barnett,
david.willie, stefanus.kurniawan<EMAIL_ADDRESS>srikanth.thudumu,
<EMAIL_ADDRESS>Applied Artificial Intelligence Institute,
Deakin UniversityGeelongAustralia

###### Abstract.

Large language models (LLMs) enable state-of-the-art semantic capabilities to
be added to software systems such as semantic search of unstructured documents
and text generation. However, these models are computationally expensive. At
scale, the cost of serving thousands of users increases massively affecting
also user experience. To address this problem, semantic caches are used to
check for answers to similar queries (that may have been phrased differently)
without hitting the LLM service. Due to the nature of these semantic cache
techniques that rely on query embeddings, there is a high chance of errors
impacting user confidence in the system. Adopting semantic cache techniques
usually requires testing the effectiveness of a semantic cache (accurate cache
hits and misses) which requires a labelled test set of similar queries and
responses which is often unavailable. In this paper, we present VaryGen, an
approach for using LLMs for test input generation that produces similar
questions from unstructured text documents. Our novel approach uses the
reasoning capabilities of LLMs to 1) adapt queries to the domain, 2)
synthesise subtle variations to queries, and 3) evaluate the synthesised test
dataset. We evaluated our approach in the domain of a student question and
answer system by qualitatively analysing 100 generated queries and result
pairs, and conducting an empirical case study with an open source semantic
cache. Our results show that query pairs satisfy human expectations of
similarity and our generated data demonstrates failure cases of a semantic
cache. Additionally, we also evaluate our approach on Qasper dataset. This
work is an important first step into test input generation for semantic
applications and presents considerations for practitioners when calibrating a
semantic cache.

Large Language Model, Query Evaluation, Question Answering, Semantic Cache,
Test Input Generation

††conference: 3rd International Conference on AI Engineering — Software
Engineering for AI; April 2024; Lisbon, Portugal††ccs: Software and its
engineering Empirical software validation

##  1\. Introduction

Software development kits for creating applications that use large language
models (LLMs) are starting to emerge i.e Microsoft’s Semantic
Kernel111https://learn.microsoft.com/en-us/semantic-kernel/overview/ and
Langchain 222https://python.langchain.com/docs/get_started/introduction. These
SDKs are growing in popularity as LLMs are able to 1) resolve ambiguities in
natural language queries, 2) reason over unstructured documents, and 3)
synthesise a human like responses to prompts (Bang, 2023). Semantic caches are
becoming an important aspect of these SDKs as developers seek to create
applications with a) lower latency, b) lower costs, and c) reduced
environmental impact from using less compute resources (i.e. no repeat calls
to LLMs with billions of parameters).

A semantic cache enables similar queries (as defined by a similarity measure
rather than an exact match) to match cached responses. However, testing the
robustness of a semantic cache is challenging due to a) application specific
calibration to balance the cache hits/misses ratio, b) domain specific
query/response pairs, and c) a suitable similarity measure that handles the
nuance in natural language (see footnote 4). Ensuring the robustness of an
application with a semantic cache relies on a representative sample of
query/response pairs for calibration and testing.

Query |  Do I have to reference in my Assignment 2?
---|---
Cached Query |  In Assignment 2, how many university references should I use?
Table 1. An example of an incorrect cache hit using the semantic cache
GPTCache444https://github.com/zilliztech/GPTCache.

LLMs have been used to generate data for software applications (Li et al.,
2023; Wang et al., 2023; Kang et al., 2023; Liu et al., 2023; Alonso et al.,
2022; Mariani et al., 2014) and for testing information retrieval based
applications (Alaofi et al., 2023). They base their work on the assumption of
pre-existing questions which for new applications is not available. Recent
work has also explored the use of LLMs for generating data for training Q&A
systems. However, their work focuses on the volume of questions from documents
rather than subtle variations that would expose the limits of a semantic
caching strategy. In summary, prior work has not considered the problem of
test input generation for testing LLM based Q&A systems. Our goal is to
improve the robustness of Q&A systems by test input generation and
demonstrating the effectiveness of finding limitations in a semantic cache.

In this paper, we propose an innovative approach that uses LLMs for generating
domain specific query/response pairs. We call this approach as Variation
Generator (VaryGen). VaryGen uses LLMs to a) generate queries, b) create hard
negatives, and c) verify the quality of the generated pairs. We take a two
stage approach, first we synthesise domain specific questions from a
collection of documents, optionally including domain specific terms in the
generation process. Second, from the synthesised questions we create
variations resulting in hard negatives and close matches. To evaluate our
approach, we 1) conducted a qualitative analysis of the synthesised questions
(N=100), and 2) through a case study of an open source semantic cache,
GPTCache. Our results found limitations in the default configurations of
GPTCache demonstrating that our approach is suitable for test input
generation. However, the qualitative evaluation revealed additional filters
are required to improve the variations of questions which we leave for future
work.

The contributions arising from this work are:

  * •

An automated approach for synthesising query/response pairs that contain hard
negatives. This approach has been designed for testing applications that
include QandA or a semantic cache for natural language queries.

  * •

A qualitative analysis of the challenges with using LLMs for synthesising test
input data.

  * •

An empirical case study with an open source semantic cache, GPTCache. This
showed the nuances in similarity measures that software engineers need to test
to improve the robustness of their applications.

Figure 1. The 3 stages in VaryGen : 1) Query Generation, 2) Query Evaluation
and 3) Query Variation Generation. Green boxes indicate steps that involve a
LLM and blue boxes that are automated without the use of an LLM.

##  2\. Semantic Cache

For the purpose of this paper, we investigate the use of a semantic cache as
part of a question and answering system where the queries are questions and
the answers are the responses. Semantic caching has been a long studied
problem (Godfrey and Gryz, 1999; Ahmad et al., 2020; Ajarroud and Zellou,
2022; Salas and Hogan, 2022; Frieder et al., 2022) for efficient retrieval of
query responses by reusing already extracted data. A semantic cache is
designed to optimise the performance and efficiency of systems that process
large volumes of data, such as those based on LLMs. Unlike traditional caches
that store data based on the exact queries or data requests, a semantic cache
understands the meaning or ”semantics” of the data and the queries. This
allows it to not just store exact query results, but also data that is
semantically related to these queries. The semantics are represented by a text
embedding, an array of floating point values that has been trained with a
language model. When a new query is made, the semantic cache retrieves data
that is contextually relevant, even if the new query is not an exact match to
previous ones.

Text embeddings are imprecise for subtle variations in natural language and
for domain specific terminology. When using a semantic cache, a search
algorithm such as K-nearest neighbours is used to find relevant embeddings
before filtering to a single result. This approach requires calibration as it
will always return a result even if the answers aren’t similar. The robustness
of an LLM based application requires a well calibrated semantic cache.
Incorrect configuration results in incorrect cache hits (i.e. the user
receives an incorrect answer), or many incorrect cache misses - mitigating the
benefits of using a semantic cache. Each application requires calibration due
to domain specific terminology and diversity of content i.e. the quality of
the embeddings for each domain is unknown until tested. To calibrate a
semantic cache a representative dataset is required that includes both true
matches and almost but incorrect matches.

Thus a test input generation approach for semantic caches requires the
following:

  * •

Generate domain relevant queries automatically.

  * •

Include a test oracle to ensure that generated queries are relevant to the
domain.

  * •

Create semantically ”close” variations of queries (hard negatives and true
pairs).

  * •

Automated to scale the data generating process to support accurate
calibration.

##  3\. VaryGen

Based on the discussion above, we propose a test input generation approach
using LLM. This 3-step approach, (detailed below) depicted in Figure 1,
combines the strengths of LLMs, domain knowledge, and rigorous evaluation to
generate diverse, contextually relevant, and high-quality test input data for
semantic applications.

###  3.1. Query And Answer Synthesis

In formulating our approach, we made three assumptions: 1) a corpus of
documents is available, 2) the documents are a semantic representation (i.e.
an embedding vector that represents the semantic content of the document), and
3) a way to retrieve answers to queries from the documents is available.

The initial step involves extracting relevant answers from the provided
document. The process of answer extraction from documents for test input
generation involves utilizing LLMs and a document corpus and is as follows:

First, we provide a document to an LLM and prompt it to extract facts from the
document. Most documents consist of multiple facts or sentences that can serve
as an answer to specific questions. Thus, we use these facts as answers for
generating questions. Note that, if the document is too long or consists of
multiple documents, a retriever can be used to extract relevant facts from it.
However, for simplicity, we provide full document passages in the prompt to an
LLM to extract answers. We then again prompt the LLM to generate questions for
each of the extracted answer. Here, proper prompting and using extracted
answers restrict the generated questions to the scope of the document only,
thereby obtaining domain relevant queries automatically.

###  3.2. Query Evaluation

The generated queries obtained in the previous step need to be verified for
correctness and whether they align contextually with the content of the
document, which is the aim of this step.

This step validates the accuracy of the generated queries by mapping them back
to the original document. For each query, the top $N$ documents are retrieved
and assessed whether the original document is included in the returned set.
The queries that extract relevant documents are included while the others are
excluded from the set. The refined set of generated queries is then used in
the next step to generate variations of the queries.

Thus, this verification step ensures that the obtained queries align
contextually with the content of the document (i.e., they are domain
specific), establishing a reliable connection between the generated questions
and the source material.

Question: {question}
---
Answer: {answer}
Generate variations ONLY from the provided Question above and ONLY use the
Answer to constrain the generated questions.
Variation Guidelines: {….}
Response:
Table 2. Prompt template for Query Variation Generation

###  3.3. Query Variation Generation

This step focuses on generating semantically similar variations of the queries
obtained in the previous step. The idea is to generate as diverse variations
of the query as possible while still maintaining semantic similarity.

We prompt the LLM to generate variations of the queries making sure that: (i)
the queries are diverse i.e. different in terms of phrasing, structure, or
specific keywords, and (ii) any unrelated or hallucinated queries are removed.
This maintains the quality of the generated variations. We provide both the
question and the answer in the prompt in order to keep the queries related to
the extracted content and that the domain knowledge is reflected in the
generated variations.

Utilising proper prompting, this step refines the queries by incorporating
specific terms, concepts, or structures relevant to the domain, ensuring that
the generated test inputs align closely with the intricacies of the subject
matter. Table 2 provides an example of a prompt used to achieve the above
objective. The guidelines help to provide relevance and specificity in the
query variations.

Thus, using the above approach, we get the required query variations that can
be used to test semantic applications. Next, we discuss the evaluation of our
input data generation approach.

##  4\. Preliminary Evaluation

In this section, we discuss the dataset details and the evaluation strategy
along with the results of the evaluations.

###  4.1. Dataset Description

We use two datasets for our study. The Qasper dataset (Dasigi et al., 2021) is
an information-seeking dataset consisting of research papers on natural
language processing. The dataset contains question-answers with different
question types such as abstractive, extractive, boolean and unanswerable. We
filter the dataset by removing the unanswerable ones for our purpose.

The Assignment dataset is a manually curated dataset containing 61 questions
crafted from sample questions representing what students are likely to ask
during a university subject. The dataset consists of questions regarding the
assignments such as word limit, requirements, and use of GenAI. As some
questions can be similar, we further excluded questions that are too similar
by manual inspection, leaving only 52 unique questions.

###  4.2. Qualitative Evaluation

The focus of VaryGen was to create an automated query and answer generation
pipeline to support software engineers working with a semantic cache. We
conducted evaluations in two phases to ensure a thorough analysis of the
VaryGen approach. These evaluations covering the different steps of VaryGen
ensure representative samples of each of the core stages is evaluated.

Method: The initial evaluation involved a human assessment of the generated
questions. We conducted a random sampling of 100 questions, scrutinizing each
to determine if the associated answers appeared relevant to the generated
questions. Our criterion for marking a question as ‘Correct’ included
assessing both the relevance of the answer to the question and the overall
realism of the question – whether it resembled the type of questions students
would typically ask. Questions meeting these criteria were marked as
‘Correct’, while those that did not were labelled as ‘Incorrect’. We conducted
an initial evaluation for the Assignment dataset as it consists of documents
and not for the Qasper dataset, which consisted of questions that could be
directly used for the next step.

The second phase of evaluation involved a human assessment of the query
variations generated for each query. In this stage, we randomly selected 100
question-answer pairs and examined each question’s correctness by comparing it
against the original answer. Essentially, our focus was on verifying whether
the generated questions aligned sensibly with the corresponding answer. This
evaluation is done for both datasets.

Results: The results of the initial evaluation on the Assignment dataset
indicate that 81% of the generated questions were realistic, while only 19%
did not satisfy our criteria. This demonstrates that our approach generates
relevant questions that are domain-specific and realistic. For example,
questions such as ”What do I need to formulate that is original and related to
the topic I have chosen for Assessment Task 2?” may not be asked by a student.
We now discuss the evaluation results on query variations. For the Assignment
dataset, we found that 80% of the sampled query variations were correct, while
20% were incorrect. Similarly, for the Qasper dataset, the correct was 84%
while the incorrect % was only 16.

These results from the evaluations emphasise that most generated questions and
the variations of the query for both datasets were semantically similar to the
original questions from which the variations were generated.

Strategy |  Correct Hits |  Incorrect Hits |  Correct Misses |  Incorrect Misses
---|---|---|---|---
DistilBERT |  1234 |  885 |  19 |  128
SBERT |  1422 |  360 |  38 |  446
ONNX |  1111 |  314 |  35 |  806
Table 3. Results from evaluating the generated dataset against three semantic
caching strategies

###  4.3. Semantic Cache Case Study

We analysed the performance of an open source semantic cache library using the
generated dataset.

Method. For the purpose of our case study, we evaluated GPTCache (Bang, 2023),
an open source semantic cache library with various semantic caching
strategies. GPTCache was selected due to a) intended use for LLM applications,
and b) popular (over 5000 stars). We used the Assignment dataset and evaluated
the performance of GPTCache using three strategies with a similarity threshold
of 0.9 (selected through empirical evaluation):

  * •

DistilBERT (base, uncased) embeddings with Cosine Similarity for similarity
evaluation

  * •

SBERT embeddings with GPTCache’s SBERT CrossEncoder implementation for
similarity evaluation

  * •

ONNX embeddings with GPTCache’s ALBERT ONNX implementation for similarity
evaluation

The following metrics were used to measure the performance of GPTCache: Number
of correct cache hits, Number of incorrect cache hits, Number of correct cache
misses and Number of incorrect cache misses. Correct Hits indicate a correct
value was retrieved from the cache. Incorrect Hits indicate a non-matching
value was returned. Correct Misses indicate no value was returned when no
value was expected, and Incorrect Misses indicate a value was not returned
where one was expected.

Results. We used the Assignment dataset consisting of a total of 2266
questions including the original questions and their variations generated
using our approach. We put the dataset of questions into the cache, one by
one, and measure the above metrics. The results of our evaluation are shown in
Table 3 which demonstrates the generated question set was capable of providing
examples that resulted in Correct hits, Incorrect Hits and Incorrect Misses on
all three caching strategies.

Strategy |  Query |  Cached Key
---|---|---
DistilBERT |  For the success of this unit, is the use of generative AI necessary? |  Does this unit require to use generative AI?
SBERT |  In Assignment 2, how many academic citations are necessary? |  How many scholarly references should be included in Assessment 2?
ONNX |  How extensive should my description of using generative AI be for Assessment 1? |  How comprehensive should I be when explaining my use of generative AI for Assignment 1?
Table 4. True Positive - Correct Cache Hit Strategy |  Query |  Cached Key
---|---|---
DistilBERT |  What is the word limit for the background section of my Assessment 1? |  What is the recommended word count for the aims section of my report for AT1?
SBERT |  What is the directive for Assignment 1? |  How many words should go into my campaign strategy for Assignment 1?
ONNX |  How should I proceed with Assessment 2? |  How can I get a high distinction in Assessment 2?
Table 5. False Positive - Incorrect Cache Hit Strategy |  Query |  Expected Key
---|---|---
DistilBERT |  What should the layout of my AT1 look like? |  What sections should I have in my AT1?
SBERT |  What’s the brief for Assignment 1? |  What do I need to do for Assignment 1?
ONNX |  What methodologies are feasible for Assignment 2? |  What methods can I use for Assignment 2?
Table 6. False Negatives - Incorrect Cache Miss

We discuss some sample examples of the above results. For correct cache hits,
as shown in Table 4, all three sample query, while different in structure and
phrasing, are similar in meaning to the respective cached key, which validates
the relevance and accuracy of the synthesized questions. This highlights the
robustness and effectiveness of our test input generation approach in
capturing semantic relevance across many different formulations of queries.

Regarding incorrect cache hit examples shown in Table 5, the first example
shows that although both queries pertain to Assessment 1, they focus on
different aspects (background vs. aims), making them semantically distinct.
The reason for this incorrect cache hit could be attributed to the limitations
in the semantic cache’s ability to discern subtle variations in the meaning of
queries. The second and third examples are also semantically different in
terms of their requirements and follow similar explanations.

Regarding incorrect cache miss examples shown in Table 6 in the first example,
the semantic cache fails to identify the correct match, as it may not fully
capture the specific intent (sections and layouts) behind the generated query.
For the second example, the generated query is seeking information about the
brief or instructions for Assignment 1, while the expected cached query is a
more general inquiry about the tasks required for Assignment 1. The semantic
cache fails possibly due to the subtle differences in the queries semantics.
Similarly, for the third example, the terms ”methodologies” and ”methods” may
not have a direct one-to-one correspondence in the semantic embeddings.

Thus, our approach is capable of uncovering scenarios where the semantic cache
strategies may fail to identify relevant matches or may incorrectly retrieve
matches. This demonstrates areas for refinement in the specificity, diversity,
and coverage of the generated queries to enhance the performance of semantic
caching systems.

##  5\. Related Work

Recent studies have demonstrated the potential for leveraging LLMs to
automatically generate various test artifacts including test cases, test
inputs, and test oracles (Wang et al., 2023). Kang et al. (Kang et al., 2023)
proposed an approach for using an LLM to reproduce bugs and generate reliable
test cases to suggest to developers. Yu et al. (Yu et al., 2023) demonstrated
how well LLMs can adapt to diverse systems while creating test scripts. Liu et
al. (Liu et al., 2023) introduced an LLM-based approach to intelligently
generate semantic input text based on GUI context. Other works have focused on
using LLMs or semantic knowledge bases to generate valid test inputs. Alonso
et al. (Alonso et al., 2022) proposed ARTE, which uses knowledge bases such as
DBPedia to extract realistic web API test inputs based on API specifications.
Mariani et al. (Mariani et al., 2014) introduced Link, which analyses a GUI’s
input fields, queries DBPedia to find matching data, and generates complex
test inputs. Wanwarang et al. (Wanwarang et al., 2020) leveraged both static
and dynamic analysis along with an LLM to generate test inputs. Some studies
have investigated using LLMs to generate test oracles, which determine the
correctness of test outputs. Yang et al. (Yang et al., 2023) used an LLM to
create knowledge bases to guide testers on what outputs to validate. Alaofi et
al. (Alaofi et al., 2023) showed LLMs can generate variations of queries to
create test oracles.

LLMs show promise for the automated generation of various test artefacts,
reducing manual effort while improving test coverage and effectiveness.
However, there remain open challenges in improving the reliability and
semantic correctness of LLM-generated tests.

##  6\. Limitations and Future Research Directions

While our approach demonstrates promise in generating domain-specific
query/response pairs for testing LLMs and semantic applications, certain
limitations underscore areas for future research. The need for additional
filters to enhance the quality of query variations, identified through our
evaluation, prompts further investigation into advanced techniques for
refinement. Additionally, our approach currently lacks human validation during
data generation, which means some generated questions may be of lower quality
or incorrect. However, we argue that an automated, inexpensive dataset that
uncovers software limitations, as shown in our case study, still provides
utility. Incorporating human annotation to improve dataset quality remains an
area for future work.

Additionally, the inherent challenge of subtle semantic nuances and
imprecisions in text embeddings requires attention, urging the exploration of
methods to fine-tune LLMs or incorporate domain-specific embeddings. Our
evaluation also focused on a single semantic cache, so further empirical
comparisons of multiple semantic caching systems should be carried out, which
is now feasible using VaryGen . Other potential areas for future work include
performing ablation studies to reduce data generation costs for increased
scale, defining quality models and metrics to improve query generation,
additional empirical evaluations of open source caches, generating datasets to
produce guards protecting semantic applications, incorporating diversity
metrics to broaden the scope of questions, and further analysis of how
synthesis techniques could be tailored and improved for diverse domains and
applications.

Addressing the limitations and exploring these research directions will
contribute to the continual improvement and advancement of test input
generation methodologies for LLMs and semantic applications.

##  7\. Conclusion

This paper has explored the evolving role of software development kits (SDKs)
that leverage large language models (LLMs) to enable more advanced natural
language applications. Semantic caches have emerged as a pivotal component
within these SDKs, aiming to reduce latency, costs, and environmental impact
by avoiding redundant calls to the LLM. However, the imprecision of text
embeddings requires careful calibration of semantic caches to prevent
incorrect cache hits. To address this, we have proposed and evaluated an
innovative approach for automated test input generation using LLMs themselves.
Through a two-stage process of synthesising domain-specific queries and
creating semantically close variations, including hard negatives,
representative test data can be generated at scale. Both the qualitative
analysis and semantic cache case study validate the suitability of this
approach for the robust testing necessary to ensure cache calibration. This
research contributes a holistic perspective, not just on test input
generation, but on the end-to-end robustness and calibration needs of semantic
caches in this evolving landscape of LLM-based SDKs. The proposed approach
paves the way for enhanced software development practices that fully harness
advanced language models while safeguarding reliability.

##  8\. Acknowledgements

To Rajesh Vasa and Kon Mouzakis for their insightful guidance and unwavering
support that greatly enriched this research project.

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]January 5, 2020

#  A Classical $\pi$ Machine and Grover’s Algorithm

Jiang Liu<EMAIL_ADDRESS>
([)

###### Abstract

This paper studies a well-known $\pi$ machine illustrated by Fig. (1). It is
shown that the $\pi$ machine can compute digits of $\pi$ if the ratio of block
weights, $m_{2}/m_{1}$, satisfies certain conditions, and that dynamics of the
$\pi$ machine is identical to that of Grover’s algorithm Grover (1996) in
quantum computing.

Two-Block-Collision $\pi$ machine; Grover’s Algorithm, Quantum Computing

##  I Introduction

Computation of $\pi$ has never ceased to delight us111Current world record of
$\pi$ digits is 31.4 trillion by Emma Haruka Iwao using Chudnovsky algorithm
Chudnovsky and Chudnovsky (1988) on Google Cloud Iwao (2019); Kleinman (2019);
Shaba (2019). One intereting methodIllner (2013); Antonick (2014);
Aretxabaleta  _et al._ (2017); Fraser (2019) is shown in Fig. (1), where
digits of $\pi$ is computed by counting the number of elastic collisions
between two sliding blocks of masses $m_{1}$ and $m_{2}$, and collisions
between block $m_{1}$ and a wall. This paper is to take a fresh look at this
simple machine.

Figure 1:  Initial state of two sliding block elastic collision $\pi$ machine.

The rest of this paper is organized as follows. Section II shows that
calculability of $\pi$ digits depends on $m_{2}/m_{1}$. Section III shows that
dynamics of the $\pi$ machine is the same as that of Grover’s algorithm Grover
(1996), and that Grover’s quantum computing Nielsen and Chuang (2010) can be
visualized classically by the $\pi$ machine. Our conclusion is summarized in
Section V.

##  II The $\pi$ Machine

Consider a weighted velocity space shown by Fig. (2), where

$\bm{v}_{t}\equiv\begin{pmatrix}&\sqrt{m_{2}}v_{2,t}\\\
&\sqrt{m_{1}}v_{1,t}\end{pmatrix}$ (1)

and $v_{1,t}$ and $v_{2,t}$ are the velocities of $m_{1}$ and $m_{2}$ at $t$.

Figure 2:  Geometric representation of a $\pi$ machine’s dynamics whose
$\theta^{*}=\pi/12$. Doted lines parallel to $(0,v)-(0,-v)$ axis, represent
$m_{1}$ bouncing off the wall. Doted lines parallel to $(0,v^{*})-(0,-v^{*})$
axis represent $m_{1}-m_{2}$ collision. $\theta_{t}$ is the polar angle of
$\bm{v}_{t}$ after each $m_{1}-m_{2}$ collision.

Conservation of kinetic energy requires that $\bm{v}_{t}$ be on a circle of
radius $|\bm{v}_{t}|=\sqrt{m_{2}}v_{2,0}$. Conservation of momentum implies

$m_{2}v_{2,t+1}+m_{1}v_{1,t+1}=m_{2}v_{2,t}-m_{1}v_{1,t},$ (2)

where minus sign of $m_{1}v_{1,t}$ rises from $m_{1}$ bouncing off the wall.
In polar coordinates, where $\sqrt{m_{2}}v_{2,t}=|\bm{v}_{t}|\cos\theta_{t}$
and $\sqrt{m_{1}}v_{1,t}=|\bm{v}_{t}|\sin\theta_{t}$, Eq. (2) can be
simplified to

$\cos(\theta_{t+1}-\theta^{*})=\cos(\theta_{t}+\theta^{*}),$ (3)

where

$\sin\theta^{*}=\sqrt{\frac{m_{1}}{m_{1}+m_{2}}}.$ (4)

Solution to Eq. (3) with initial condition $\theta_{0}=\pi$ is

$\theta_{t}=2t\theta^{*}+\pi.$ (5)

Collision stops at $T$ when $v_{1,T}\to 0$, i.e., $\theta_{T}\to 2\pi$. Total
number of collisions between $t=0$ and $t=T$ is $2T$ and from Eq. (5)

$2T=\left\lfloor\frac{\pi}{\theta^{*}}\right\rfloor.$ (6)

Clearly, number of collisions prints digits of $\pi$ if $\theta^{*}=10^{-n}$,
where $n$ is a positive integer. This happens when $m_{2}/m_{1}=10^{2n}\gg 1$
(see Eq. (4)). This is the necessary condition for the calculability of the
$\pi$ machine. Otherwise, the $\pi$ machine cannot produce digits of $\pi$.

##  III The $\pi$ Machine and Grover’s Algorithm

That dynamics of the $\pi$ machine and Grover’s algorithm Grover (1996) are
identical can be viewed most conveniently from Hamiltonian formulation, in
which collisions are described by evolution of states of bases $\bm{v}$ and
$\bm{v^{*}}$.

In $\bm{v}$ basis, $\bm{v}_{t}$ is an eigenvector of Hamiltonian for $m_{1}$
bouncing off the wall. Geometrically, $\bm{v}_{t}$ reflects over the $(v,0)$
axis of Fig. (2), i.e.,

$\bm{v}_{t}\rightarrow\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\bm{v}_{t}.$ (7)

$\bm{v}$ basis is referred to as computational basis in the literature. It is
called computational due to interactions between internal system (blocks of
the $\pi$ machine) and environment(the wall).

$\bm{v^{*}}$ basis is related to $\bm{v}$ basis by a rotation (see Fig. (2))
with $\theta^{*}$ given by Eq. (4)

$\bm{v^{*}}_{t}=U_{\theta^{*}}\bm{v}_{t},$ (8)

where

$U_{\theta^{*}}=\begin{pmatrix}\cos\theta^{*}&\sin\theta^{*}\\\
-\sin\theta^{*}&\cos\theta^{*}\end{pmatrix}.$ (9)

In $\bm{v^{*}}_{t}$ basis, $\bm{v^{*}}_{t}$ is an eigenvector of Hamiltonian
for block collision. Geometrically, $\bm{v^{*}}_{t}$ reflects over the
$(v^{*},0)$ axis of Fig. (2), i.e.,

$\bm{v^{*}}_{t+1}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\bm{v^{*}}_{t}.$ (10)

$\bm{v^{*}}$ basis is referred to as canonical basis in the literature.
Canonical basis does not involve environment.

It then follows from Eqs. (7) to (10) that dynamics of the $\pi$ machine in
$\bm{v}$ basis (computational basis) is

$\bm{v}_{t+1}=G_{\theta^{*}}\bm{v}_{t},$ (11)

where

$G_{\theta^{*}}=U_{\theta^{*}}^{\dagger}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}U_{\theta^{*}}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (12)

Upper component of Eq. (11) is Eq. (3) discussed in the previous section.
Lower component of Eq. (11) is the conjugate of Eq. (3),
$\sin(\theta_{t+1}-\theta^{*})=\sin(\theta_{t}+\theta^{*})$. Both lead to the
same result.

Apart from an overall phase of no physical significance, dynamics described by
Eqs. (11) and (12) is the same as that of Grover’s diffusion operator Grover
(1996); Nielsen and Chuang (2010).

As a result, processes described by Fig. (2) for the $\pi$ machine can be
interpolated into Grover’s quantum circuits shown by Fig. (3). This
interpolation allows us to visualize Grover’s quantum computing by the $\pi$
machine.

Figure 3:  Quantum $\pi$ machine simulator (Grover’s circuit). $|W\rangle$ is
Grover’s working qubit representing the wall. $|0\rangle$ is the initial
$n$-qubit state in canonical basis representing block state $(-\bm{v},0)$.
$|k\rangle$ is the search target state in computational basis representing
block state $\sqrt{m_{1}}v_{1,0}$.

Working qubits $|W\rangle$ in Grover’s algorithm can be visualized as the
wall, searching target $|k\rangle$ as $\sqrt{m_{1}}v_{1,0}$ and superposition
of the rest of qubits as $\sqrt{m_{2}}v_{2,0}$; Total number of qubits $n$ is
determined by $n=\lceil\log_{2}(1+m_{2}/m_{1})\rceil$.

Grover’s diffusion operator $G_{\theta^{*}}$ can be visualized as $m_{1}$
bouncing off the wall in computational basis, which plays the same role as
Grover’s Oracle, i.e., $|k\rangle\to-|k\rangle$, followed by $m_{1}$ and
$m_{2}$ collision in canonical basis, which plays the same role as phase
shift, i.e., $|0\rangle\to-|0\rangle$. Number of diffusion operators can be
visualized as number of collisions.

Most remarkably, basis mixing Eq. (9), which is generated in Grover’s
algorithm by Hadamard gate rotating qubit along polar axis of Bloch
SphereBloch (1946), is a consequence of conservation of kinetic energy and
momentum of the $\pi$ machine.

There is one difference between the $\pi$ machine and Grover’s search. In the
$\pi$ machine, initial state can be prepared and final state can be measured
in computational basis alone. In Grover’s search, however, initial state
$|0\rangle$ is prepared in canonical basis and final state is measured in
computational basis. Probability of finding $|k\rangle$ of computational
basis, which is equivalent to measuring $\sqrt{m_{1}}v_{1,t}$, from initial
state $|0\rangle$ of canonical basis is

$Pr(\langle k|0\rangle)_{t}=\sin^{2}\theta_{t},$ (13)

where $\theta_{t}=2t\theta^{*}+\theta^{*}$ (mod $\pi$), which has an extra
$\theta^{*}$ when compared to Eq. (5).

Apart from the difference of initial state, dynamics of the $\pi$ machine and
Grover’s algorithm are identical. Computational complexity of these two
processes is the same. In order words, the classical $\pi$ machine is
effectively simulating a type of Grover’s quantum computing.

##  IV Conclusion

Results of this paper can be summarized as follows: digits of $\pi$ can be
computed by the $\pi$ machine only in limited cases. Dynamics of the $\pi$
machine is the same as that of Grover’s algorithm in all cases. Grover’s
quantum computing may be classically visualized by the $\pi$ machine.

After the completion of this work, I noticed from Quanta Magazine Sanderson
(2020) a very interesting recent work of Adam Brown Brown (2019) on the same
subject. Although our approaches are somewhat different, we independently
reached the same conclusion.

###### Acknowledgements.

I would like to thank Ken Li for valuable discussions. He verified numerically
the validity of Eqs. (4) and (6). Any views expressed are mine as an
individual and not as a representative speaking for or on behalf of my
employer, nor do they represent my employer’s positions, strategies or
opinions. This research did not receive any specific grant from funding
agencies in the public, commercial, or not-for-profit sectors. Declarations of
interest: None.

## References

  * Grover (1996) L. K. Grover, Proceedings of 28th Annual ACM Symposium on the Theory of Computing , 212 (1996).
  * Chudnovsky and Chudnovsky (1988) D. Chudnovsky and G. Chudnovsky, Approximation and complex multiplication according to ramanujan, Ramanujan revisited: Proceedings of the Centenary Conference (1988).
  * Iwao (2019) E. H. Iwao, Most accurate value of $\pi$, Guinness World Records Retrieved 2019-03-14 (2019).
  * Kleinman (2019) Z. Kleinman, Emma Haruka Iwao smashes pi world record with Google help, BBC News, 2019-03-14 (2019).
  * Shaba (2019) H. Shaba, A Google employee just shattered the record for Pi calculation. Her name is Emma Haruka Iwao, washingtonpost.com. Washington Post 2019-03-14 (2019).
  * Illner (2013) R. Illner, Hidden Circles and the Digits of $\pi$, PI in the SKY (2013).
  * Antonick (2014) G. Antonick, The Pi Machine, https://wordplay.blogs.nytimes.com/2014/03/10/pi/ (2014).
  * Aretxabaleta  _et al._ (2017) X. M. Aretxabaleta, M. Gonchenko, N. Harshman, S. G. Jackson, M. Olshanii, and G. E. Astrakharchik, Two-ball billiard predicts digits of the number PI in non-integer numerical basees, arXiv: 1712.06698v1 [math.DS] (2017).
  * Fraser (2019) C. Fraser<EMAIL_ADDRESS>2019-01-19 (2019).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2010).
  * Bloch (1946) F. Bloch, Phys. Rev. 70, 1946 (1946).
  * Sanderson (2020) G. Sanderson, How Pi Connects Colliding Blocks to a Quantum Search Algorithm, Quanta Magazine, 21 Jan (2020).
  * Brown (2019) A. R. Brown, Playing Pool with $|\psi\rangle$: from Bouncing Billiards to Quantum Search, arXiv:1912.02207v1 [quant-ph], 4 Dec (2019).


# A Gleason–Kahane–Żelazko theorem for reproducing kernel Hilbert spaces

Cheng Chu  Département de mathématiques et de statistique, Université Laval,
Québec City (Québec), Canada G1V 0A6<EMAIL_ADDRESS>,  Michael Hartz
Fachrichtung Mathematik, Universität des Saarlandes, 66123 Saarbrücken,
Germany<EMAIL_ADDRESS>,  Javad Mashreghi  Département de mathématiques
et de statistique, Université Laval, Québec City (Québec), Canada G1V 0A6.
<EMAIL_ADDRESS>and  Thomas Ransford  Département de
mathématiques et de statistique, Université Laval, Québec City (Québec),
Canada G1V 0A6<EMAIL_ADDRESS>
(Date: 20 Aug 2021)

###### Abstract.

We establish the following Hilbert-space analogue of the Gleason–Kahane-
Żelazko theorem. If ${\mathcal{H}}$ is a reproducing kernel Hilbert space with
a normalized complete Pick kernel, and if $\Lambda$ is a linear functional on
${\mathcal{H}}$ such that $\Lambda(1)=1$ and $\Lambda(f)\neq 0$ for all cyclic
functions $f\in{\mathcal{H}}$, then $\Lambda$ is multiplicative, in the sense
that $\Lambda(fg)=\Lambda(f)\Lambda(g)$ for all $f,g\in{\mathcal{H}}$ such
that $fg\in{\mathcal{H}}$. Moreover $\Lambda$ is automatically continuous. We
give examples to show that the theorem fails if the hypothesis of a complete
Pick kernel is omitted. We also discuss conditions under which $\Lambda$ has
to be a point evaluation.

###### Key words and phrases:

Reproducing kernel Hilbert space, complete Pick kernel, multiplier, cyclic
function, point evaluation, automatic continuity

###### 2010 Mathematics Subject Classification:

46E22, 46H40

##  1\. Introduction and statement of main result

The following result, known as the Gleason–Kahane–Żelazko (GKZ) theorem,
characterizes multiplicativity of a linear functional on a Banach algebra.

######  Theorem 1.1 ([11, 15, 24]).

Let $A$ be a complex unital Banach algebra, and let $\Lambda:A\to{\mathbb{C}}$
be a linear functional such that $\Lambda\not\equiv 0$. The following
statements are equivalent:

  1. (i)

$\Lambda(1)=1$ and $\Lambda(a)\neq 0$ for all invertible elements $a\in A$.

  2. (ii)

$\Lambda(ab)=\Lambda(a)\Lambda(b)$ for all $a,b\in A$.

Continuity of the functional is not assumed, but it is a consequence of the
theorem, since characters on a Banach algebra are always continuous.

Analogues of this result for certain holomorphic function spaces that are not
algebras were obtained in [16, 17]. Our goal in this paper is to establish the
following theorem, which is a version for abstract function spaces.

######  Theorem 1.2.

Let ${\mathcal{H}}$ be a reproducing kernel Hilbert space with a normalized
complete Pick kernel, and let $\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ be a
linear functional such that $\Lambda\not\equiv 0$. The following statements
are equivalent:

  1. (i)

$\Lambda(1)=1$ and $\Lambda(f)\neq 0$ for all cyclic elements
$f\in{\mathcal{H}}$.

  2. (ii)

$\Lambda(fg)=\Lambda(f)\Lambda(g)$ for all $f,g\in{\mathcal{H}}$ such that
$fg\in{\mathcal{H}}$.

Moreover, a functional $\Lambda$ satisfying (i) or (ii) is automatically
continuous.

The terminology is explained in §2, the automatic continuity of $\Lambda$ is
established in §3, and the rest of the theorem proved in §4. In §5 we present
examples showing that the theorem may fail if we drop the assumption that
${\mathcal{H}}$ has a complete Pick kernel. Finally, in §6 we discuss
conditions under which we may conclude that $\Lambda$ is a point evaluation at
a point of $X$.

##  2\. Background on reproducing kernel Hilbert spaces

In this section we give a very brief introduction to reproducing kernel
Hilbert spaces and complete Pick kernels. For more details we refer to the
books of Paulsen–Raghupathi [19] and Agler–McCarthy [3].

###  2.1. Reproducing kernel Hilbert spaces

A _reproducing kernel Hilbert space_ (RKHS for short) is a complex Hilbert
space ${\mathcal{H}}$ of functions on a set $X$ with the property that, for
each $x\in X$, the evaluation functional $f\mapsto f(x)$ is continuous on
${\mathcal{H}}$. By the Riesz representation theorem, there is a uniquely
determined element $k_{x}\in{\mathcal{H}}$ such that $f(x)=\langle
f,k_{x}\rangle$ for all $f\in{\mathcal{H}}$. The _reproducing kernel_
associated to ${\mathcal{H}}$ is the function $K:X\times X\to{\mathbb{C}}$
defined by $K(x,y):=k_{y}(x)=\langle k_{y},k_{x}\rangle$.

A kernel $K$ is automatically a positive semi-definite function. This means
that, for every finite subset $\\{x_{1},x_{2},\dots,x_{n}\\}$ of $X$, the
$n\times n$ matrix $(K(x_{i},x_{j}))$ is positive semi-definite.

We shall always assume that $K(x,x)>0$ for all $x\in X$. This is equivalent to
supposing that, for each $x\in X$, there exists $f\in{\mathcal{H}}$ such that
$f(x)\neq 0$. In particular, this is the case if $K$ is normalized (see §2.4
below).

###  2.2. Multipliers

Let ${\mathcal{H}}$ be a RKHS on $X$. A _multiplier_ of ${\mathcal{H}}$ is a
function $h:X\to{\mathbb{C}}$ such that $hf\in{\mathcal{H}}$ whenever
$f\in{\mathcal{H}}$. We denote by ${\mathcal{M}}$ the set of multipliers. By
the closed graph theorem, if $h\in{\mathcal{M}}$, then $M_{h}:f\mapsto hf$ is
a bounded linear operator on ${\mathcal{H}}$. We define $\|h\|_{\mathcal{M}}$
to be the operator norm of $M_{h}$. With this norm ${\mathcal{M}}$ becomes a
Banach algebra.

If $h\in{\mathcal{M}}$ and $x\in X$, then a standard calculation shows that
$M_{h}^{*}k_{x}=\overline{h(x)}k_{x}$, so $\overline{h(x)}$ is an eigenvalue
of $M_{h}^{*}$, and $|h(x)|\leq\|h\|_{\mathcal{M}}$. Consequently, every
multiplier $h$ is a bounded function on $X$ with
$\sup_{X}|h|\leq\|h\|_{\mathcal{M}}$.

###  2.3. Complete Pick kernels

Let $K$ be the reproducing kernel of a RKHS ${\mathcal{H}}$ on $X$. It is a
_complete Pick kernel_ if $K(x,y)\neq 0$ for all $x,y\in X$ and if there
exists $x_{0}\in X$ such that

(1) $F(x,y):=1-\frac{K(x,x_{0})K(x_{0},y)}{K(x,y)K(x_{0},x_{0})}$

is a positive semi-definite function on $X\times X$. If this condition holds
for one $x_{0}\in X$, then it automatically holds for all $x_{0}\in X$.

Complete Pick kernels are so called because of their relation with the
solution of Pick interpolation problems. This is described in detail in [3].
Examples of RKHS with complete Pick kernels include the Hardy space $H^{2}$,
the Drury–Arveson spaces $H^{2}_{d}$ [3], the classical Dirichlet space
${\mathcal{D}}$ [1], all Dirichlet spaces with superharmonic weights
${\mathcal{D}}_{w}$ [23], the Sobolev space $W^{2}_{1}(0,1)$ [20], as well as
certain de Branges–Rovnyak spaces ${\mathcal{H}}(b)$ [8].

###  2.4. Normalized kernels

A reproducing kernel $K$ for a RKHS ${\mathcal{H}}$ on $X$ is said to be
_normalized_ if there exists $x_{0}\in X$ such that $K(x,x_{0})=1$ for all
$x\in X$. In this case $k_{x_{0}}(x):=K(x,x_{0})=1$ for all $x\in X$. Since
$k_{x_{0}}\in{\mathcal{H}}$, it follows that ${\mathcal{H}}$ contains the
constant functions, and hence that ${\mathcal{M}}\subset{\mathcal{H}}$.

We require the following well-known fact, which follows for instance from the
discussion surrounding Equation (2.2) in [12]. For the convenience of the
reader, we provide a short proof.

######  Proposition 2.1.

Let ${\mathcal{H}}$ be a RKHS on $X$ with a normalized complete Pick kernel
$K$ and let ${\mathcal{M}}$ be its multiplier algebra. Then
$k_{y}=K(\cdot,y)\in{\mathcal{M}}$ for all $y\in X$. In particular,
${\mathcal{M}}$ is dense in ${\mathcal{H}}$.

###### Proof.

Since $K$ is a normalized complete Pick kernel, a theorem of Agler and
McCarthy [2, Theorem 4.2] shows that there exists an auxiliary Hilbert space
$\mathcal{E}$ and a function $b:X\to B(\mathcal{E},{\mathbb{C}})$ with
$\|b(x)\|<1$ for all $x\in X$ such that

$K(x,y)=\frac{1}{1-b(x)b(y)^{*}}\quad(x,y\in X).$

Indeed, this follows by choosing $x_{0}$ in Equation (1) to be the
normalization point, defining $\mathcal{E}$ to be the RKHS with reproducing
kernel $F$ and $b(x)$ to be the functional on $\mathcal{E}$ of evaluation at
$x$.

Since

$(x,y)\mapsto(1-b(x)b(y)^{*})K(x,y)=1$

is a positive semi-definite function on $X\times X$, the function $b$ is a
multiplier from ${\mathcal{H}}\otimes\mathcal{E}$ into ${\mathcal{H}}$ of norm
at most $1$ (see e.g. [19, Theorem 6.28]). From this, we deduce that for each
$y\in X$, the scalar-valued function $x\mapsto b(x)b(y)^{*}$ belongs to
${\mathcal{M}}$ and

$\|b(\cdot)b(y)^{*}\|_{{\mathcal{M}}}\leq\|b(y)^{*}\|_{\mathcal{E}}<1.$

Therefore, the series

$K(\cdot,y)=\sum_{n=0}^{\infty}(b(\cdot)b(y)^{*})^{n}$

converges absolutely in the Banach algebra ${\mathcal{M}}$. Hence
$K(\cdot,y)\in{\mathcal{M}}$ for all $y\in X$. Since the linear span of the
kernel functions is dense in any RKHS, it follows in particular that
${\mathcal{M}}$ is dense in ${\mathcal{H}}$. ∎

###  2.5. Cyclic functions

Let ${\mathcal{H}}$ be a RKHS on $X$ with multiplier algebra ${\mathcal{M}}$.
Given $f\in{\mathcal{H}}$, we write $[f]:=\overline{{\mathcal{M}}f}$, the
closed ${\mathcal{M}}$-invariant subspace generated by $f$. Note that, if
$h\in{\mathcal{M}}$, then $h[f]\subset[hf]$. We say $f$ is _cyclic_ if
$[f]={\mathcal{H}}$. Clearly, if $f\in{\mathcal{H}}$ is cyclic, then $f(x)\neq
0$ for all $x\in X$. In general, the reverse is not true.

It should be remarked that some authors (notably those in [4]) define a
multiplier $h\in{\mathcal{M}}$ to be ‘cyclic’ if
$\overline{h{\mathcal{H}}}={\mathcal{H}}$. If ${\mathcal{M}}$ is dense in
${\mathcal{H}}$, then the two notions of cyclic coincide for all
$h\in{\mathcal{M}}$. In particular, this is the case whenever ${\mathcal{H}}$
has a normalized complete Pick kernel, by Proposition 2.1.

Cyclic functions play a role in RKHS analogous to that of invertible elements
in algebras. In the case of the Hardy space $H^{2}$, a function is cyclic if
and only if it is an outer function. This is a consequence of Beurling’s
classification of the closed, shift-invariant subspaces of $H^{2}$. In many
other spaces, it is difficult to characterize which functions are cyclic. In
particular, in the case of the classical Dirichlet space, this problem is the
subject of a well-known conjecture of Brown and Shields [7].

##  3\. Automatic continuity

As remarked in the introduction, it is well known (and easy to prove) that a
multiplicative linear functional on a unital Banach algebra is automatically
continuous. The corresponding result for a RKHS with a normalized, complete
Pick kernel is also true, but not quite so obvious. In fact, it is even true
under the weaker multiplicativity hypothesis that
$\Lambda(hf)=\Lambda(h)\Lambda(f)$ for all $f\in{\mathcal{H}}$ and all
$h\in{\mathcal{M}}$. As it will actually play a role in our proof of Theorem
1.2, we stop here to prove it first.

######  Theorem 3.1.

Let ${\mathcal{H}}$ be a RKHS with a normalized complete Pick kernel, let
${\mathcal{M}}$ be its multiplier algebra, and let
$\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ be a linear functional such that
$\Lambda(hf)=\Lambda(h)\Lambda(f)$ for all $f\in{\mathcal{H}}$ and all
$h\in{\mathcal{M}}$. Then $\Lambda$ is continuous on ${\mathcal{H}}$.

The proof of Theorem 3.1 hinges on the following factorization theorem due to
Jury and Martin [14, Theorem 1.1]. This result was proved by using non-
commutative factorization theorems due to Arias and Popescu [6] and Davidson
and Pitts [10].

######  Theorem 3.2.

Let ${\mathcal{H}}$ be a RKHS with a normalized complete Pick kernel, and let
${\mathcal{M}}$ be its multiplier algebra. If $(f_{n})$ is a sequence in
${\mathcal{H}}$ such that $\sum_{n}\|f_{n}\|_{\mathcal{H}}^{2}<\infty$, then
there exists a sequence $(h_{n})$ in ${\mathcal{M}}$ and a cyclic function
$g\in{\mathcal{H}}$ such that:

  * •

$f_{n}=h_{n}g$ for all $n$;

  * •

$\sum_{n}\|f_{n}\|_{{\mathcal{H}}}^{2}=\|g\|_{{\mathcal{H}}}^{2}$;

  * •

$\sum_{n}\|h_{n}f\|_{{\mathcal{H}}}^{2}\leq\|f\|_{{\mathcal{H}}}^{2}$ for all
$f\in{\mathcal{H}}$, and in particular $\|h_{n}\|_{\mathcal{M}}\leq 1$ for all
$n$.

###### Proof of Theorem 3.1.

Suppose, if possible, that $\Lambda$ is not continuous on ${\mathcal{H}}$.
Then there exists a sequence $(f_{n})$ in ${\mathcal{H}}$ such that
$\sum_{n}\|f_{n}\|_{{\mathcal{H}}}^{2}<\infty$ but
$|\Lambda(f_{n})|\to\infty$. By Theorem 3.2, we can factorize $f_{n}$ as
$f_{n}=h_{n}g$, where $g\in{\mathcal{H}}$ and $(h_{n})$ is a sequence in the
multiplier algebra ${\mathcal{M}}$ of ${\mathcal{H}}$ such that
$\|h_{n}\|_{\mathcal{M}}\leq 1$ for all $n$. By the multiplicativity
hypothesis on $\Lambda$, we have $\Lambda(f_{n})=\Lambda(h_{n})\Lambda(g)$ for
all $n$, which forces $|\Lambda(h_{n})|\to\infty$. On the other hand,
$\Lambda|_{\mathcal{M}}$ is a character on the Banach algebra ${\mathcal{M}}$,
so it is automatically continuous on ${\mathcal{M}}$, and as the sequence
$(h_{n})$ is bounded in the norm of ${\mathcal{M}}$, it follows that
$\Lambda(h_{n})$ is bounded too. We have arrived at a contradiction. ∎

##  4\. Proof of Theorem 1.2

The proof of Theorem 1.2 is based on the following result, which is a version
of the classical GKZ theorem for modules [17, Theorem 1.2].

######  Theorem 4.1.

Let $A$ be a complex unital Banach algebra, let $M$ be a left $A$-module, and
let $S$ be a non-empty subset of $M$ satisfying the following conditions:

  * (S1)

$S$ generates $M$ as an $A$-module;

  * (S2)

if $a\in A$ is invertible and $s\in S$, then $as\in S$;

  * (S3)

for all $s_{1},s_{2}\in S$, there exist $a_{1},a_{2}\in A$ such that
$a_{j}S\subset S~{}(j=1,2)$ and $a_{1}s_{1}=a_{2}s_{2}$.

Let $\Lambda:M\to{\mathbb{C}}$ be a linear functional such that
$\Lambda(s)\neq 0$ for all $s\in S$. Then there exists a unique character
$\chi$ on $A$ such that

(2) $\Lambda(am)=\chi(a)\Lambda(m)\qquad(a\in A,~{}m\in M).$

We also need another factorization theorem for RKHS’s, this one due to Aleman,
McCarthy, Richter and the second author [4, Lemma 2.3 and Theorem 3.1]. See
also §2.5 regarding the definition of cyclicity used in [4].

######  Theorem 4.2.

Let ${\mathcal{H}}$ be a RKHS on $X$ with a normalized complete Pick kernel,
and let ${\mathcal{M}}$ be its multiplier algebra. Then, given
$f\in{\mathcal{H}}$, there exist $h_{1},h_{2}\in{\mathcal{M}}$, with $h_{2}$
cyclic in ${\mathcal{H}}$, such that $f=h_{1}/h_{2}$.

###### Proof of Theorem 1.2.

We begin with the implication (i)$\Rightarrow$(ii). Assume that (i) holds,
namely $\Lambda(1)=1$ and $\Lambda(f)\neq 0$ for all cyclic functions
$f\in{\mathcal{H}}$.

The plan is to apply Theorem 4.1 with $M={\mathcal{H}}$, taking $A$ to be the
multiplier algebra ${\mathcal{M}}$ of ${\mathcal{H}}$, and $S$ to be the set
of cyclic functions in ${\mathcal{H}}$. We verify the three conditions in that
theorem:

  * (S1)

Given $f\in{\mathcal{H}}$, Theorem 3.2 implies that we can write $f=hg$, where
$h\in{\mathcal{M}}$ and $g\in S$.

  * (S2)

If $h$ is invertible in ${\mathcal{M}}$, then
${\mathcal{M}}(hf)={\mathcal{M}}f$. In particular, if $f$ is cyclic, so is
$hf$.

  * (S3)

Given $f_{1},f_{2}\in S$, Theorem 3.2 shows that we can write $f_{1}=a_{2}g$
and $f_{2}=a_{1}g$, where $a_{1},a_{2}\in{\mathcal{M}}$ and $g\in S$. Clearly
we have $a_{1}f_{1}=a_{2}f_{2}$. It remains to check that $a_{j}S\subset S$.
Let $f\in S$. Then $[a_{1}f]\supset a_{1}[f]=a_{1}{\mathcal{H}}\ni
a_{1}g=f_{2}$, and so $[a_{1}f]\supset[f_{2}]={\mathcal{H}}$. Hence $a_{1}f\in
S$. Likewise for $a_{2}$.

By Theorem 4.1, there exists a character $\chi$ on ${\mathcal{M}}$ such that
(2) holds, i.e.,

$\Lambda(hf)=\chi(h)\Lambda(f)\quad(h\in{\mathcal{M}},~{}f\in{\mathcal{H}}).$

In particular, taking $f=1$, we see that $\chi(h)=\Lambda(h)$ for all
$h\in{\mathcal{M}}$. It follows that

(3)
$\Lambda(hf)=\Lambda(h)\Lambda(f)\quad(h\in{\mathcal{M}},~{}f\in{\mathcal{H}}).$

This is nearly the desired conclusion (ii). To get the full conclusion, we
employ a trick borrowed from [4]. Let $f,g\in{\mathcal{H}}$ be such that
$fg\in{\mathcal{H}}$. By Theorem 4.2, we can write $f=h_{1}/h_{2}$, where
$h_{1},h_{2}\in{\mathcal{M}}$ and $h_{2}$ is cyclic in ${\mathcal{H}}$. Then,
using (3) several times, we have

$\displaystyle\Lambda(h_{2})\Lambda(fg)$
$\displaystyle=\Lambda(h_{2}fg)=\Lambda(h_{1}g)=\Lambda(h_{1})\Lambda(g)$
$\displaystyle=\Lambda(h_{2}f)\Lambda(g)=\Lambda(h_{2})\Lambda(f)\Lambda(g).$

Since $h_{2}$ is cyclic, we have $\Lambda(h_{2})\neq 0$, so we can cancel off
the $\Lambda(h_{2})$ terms and conclude that
$\Lambda(fg)=\Lambda(f)\Lambda(g)$, as desired. This completes the proof that
(i)$\Rightarrow$(ii).

We now turn to the reverse implication in Theorem 1.2, namely
(ii)$\Rightarrow$(i). Assume that (ii) holds, i.e.,
$\Lambda(fg)=\Lambda(f)\Lambda(g)$ for all $f,g\in{\mathcal{H}}$ with
$fg\in{\mathcal{H}}$.

Since $\Lambda(1)=\Lambda(1\cdot 1)=\Lambda(1)^{2}$, we have $\Lambda(1)=0$ or
$1$. In the former case, we have $\Lambda(f)=\Lambda(1\cdot
f)=\Lambda(1)\Lambda(f)=0$ for all $f\in{\mathcal{H}}$, contrary to the
hypothesis that $\Lambda\not\equiv 0$. So $\Lambda(1)=1$.

Let $f$ be a cyclic function in ${\mathcal{H}}$. From the definition of
cyclicity, there exists a sequence of multipliers $(h_{n})$ such that
$h_{n}f\to 1$ in ${\mathcal{H}}$. By Theorem 3.1, the assumption that
$\Lambda$ is multiplicative implies that $\Lambda$ is continuous on
${\mathcal{H}}$, and so
$\Lambda(h_{n})\Lambda(f)=\Lambda(h_{n}f)\to\Lambda(1)=1$. This implies that
$\Lambda(f)\neq 0$, and completes the proof that (ii)$\Rightarrow$(i).

Finally, the continuity statement in Theorem 1.2 follows directly from Theorem
3.1. ∎

##  5\. Two examples on the complete Pick kernel condition

In this section we present examples showing that Theorems 1.2 and 3.1 are no
longer true if we drop the assumption that ${\mathcal{H}}$ has a complete Pick
kernel. We begin with an example relating to Theorem 1.2.

######  Proposition 5.1.

There exist a RKHS ${\mathcal{H}}$ on a set $X$ and a linear functional
$\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ with the following properties:

  1. (i)

The reproducing kernel $K:X\times X\to{\mathbb{C}}$ associated to
${\mathcal{H}}$ is normalized, and satisfies $K(x,y)\neq 0$ for all $x,y\in
X$.

  2. (ii)

The linear functional $\Lambda$ satisfies $\Lambda(1)=1$ and $\Lambda(f)\neq
0$ for all cyclic elements $f\in{\mathcal{H}}$. Furthermore $\Lambda$ is
continuous on ${\mathcal{H}}$.

  3. (iii)

There exists a multiplier $h$ of ${\mathcal{H}}$ such that
$\Lambda(h^{2})\neq\Lambda(h)^{2}$.

###### Proof.

(i) Let

${\mathcal{H}}:=\Bigl{\\{}f(z_{1},z_{2}):=\sum_{j,k\geq
0}a_{jk}z_{1}^{j}z_{2}^{k}:~{}\|f\|_{\mathcal{H}}^{2}:=\sum_{j,k\geq
0}j!|a_{jk}|^{2}<\infty\Bigr{\\}}.$

This is a RKHS of holomorphic functions on ${\mathbb{C}}\times{\mathbb{D}}$.
It has the orthonormal basis $\\{z_{1}^{j}z_{2}^{k}/(j!)^{1/2}:j,k\geq 0\\}$,
from which we deduce that its reproducing kernel is given by

$K\bigl{(}(z_{1},z_{2}),(w_{1},w_{2})\bigr{)}=\sum_{j,k\geq
0}\frac{z_{1}^{j}z_{2}^{k}}{(j!)^{1/2}}\frac{\overline{w}_{1}^{j}\overline{w}_{2}^{k}}{(j!)^{1/2}}=\frac{e^{z_{1}\overline{w}_{1}}}{1-z_{2}\overline{w}_{2}}.$

Clearly $K$ is normalized at $(0,0)$, and it is non-zero everywhere on
${\mathbb{C}}\times{\mathbb{D}}$. (In fact ${\mathcal{H}}$ is the tensor
product of the Segal–Bargmann space and the Hardy space, and so $K$ is the
product of the kernels of these two spaces.)

(ii) Define $\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ by

$\Lambda\Bigl{(}\sum_{j,k\geq
0}a_{jk}z_{1}^{j}z_{2}^{k}\Bigr{)}:=a_{00}+a_{01}.$

This is a continuous linear functional on ${\mathcal{H}}$. Clearly
$\Lambda(1)=1$. Also, the condition that $\Lambda(f)\neq 0$ for all cyclic
$f\in{\mathcal{H}}$ is satisfied vacuously, because there are no cyclic
elements of ${\mathcal{H}}$. Indeed, every multiplier of ${\mathcal{H}}$ is a
bounded holomorphic function on ${\mathbb{C}}\times{\mathbb{D}}$, so by
Liouville’s theorem it must be independent of the first variable. This
evidently precludes the existence of cyclic functions.

(iii) Let $h(z_{1},z_{2}):=z_{2}$ . Multiplication by $h$ is an isometry on
${\mathcal{H}}$, so $h$ is certainly a multiplier. Also, we have
$\Lambda(h)=1$ and $\Lambda(h^{2})=0$, so $\Lambda(h^{2})\neq\Lambda(h)^{2}$.
∎

######  Remark.

In the example constructed in the proof of Proposition 5.1, the multiplier
algebra ${\mathcal{M}}$ is not dense in ${\mathcal{H}}$. This is no accident.
Indeed, if ${\mathcal{M}}$ is dense in ${\mathcal{H}}$, then every invertible
multiplier is a cyclic element of ${\mathcal{H}}$. Thus the classical GKZ
theorem (Theorem 1.1), applied to the restriction of $\Lambda$ to
${\mathcal{M}}$, shows that

$\Lambda(h_{1}h_{2})=\Lambda(h_{1})\Lambda(h_{2})\quad(h_{1},h_{2}\in{\mathcal{M}}),$

and if, further, $\Lambda$ is continuous on ${\mathcal{H}}$, then it follows
that

$\Lambda(hf)=\Lambda(h)\Lambda(f)\quad(h\in{\mathcal{M}},f\in{\mathcal{H}}).$

Concerning the continuity of $\Lambda$, one might also ask if the automatic
continuity result, Theorem 3.1, holds if we drop the assumption that
${\mathcal{H}}$ has a complete Pick kernel. The following example shows that
this is not the case, even if the multiplier algebra ${\mathcal{M}}$ is dense
in ${\mathcal{H}}$.

######  Proposition 5.2.

There exist a RKHS ${\mathcal{H}}$ of holomorphic functions on ${\mathbb{D}}$
and a linear functional $\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ with the
following properties:

  1. (i)

The reproducing kernel $K$ associated to ${\mathcal{H}}$ is normalized at $0$.

  2. (ii)

The multiplier algebra ${\mathcal{M}}$ of ${\mathcal{H}}$ equals $H^{\infty}$
and is dense in ${\mathcal{H}}$.

  3. (iii)

The linear functional $\Lambda$ satisfies $\Lambda(1)=1$ and
$\Lambda(hf)=\Lambda(h)\Lambda(f)$ for all $h\in{\mathcal{M}}$ and all
$f\in{\mathcal{H}}$.

  4. (iv)

$\Lambda$ is discontinuous on ${\mathcal{H}}$.

###### Proof.

Let $(a_{n})_{n=0}^{\infty}$ be a sequence of strictly positive real numbers
with the following properties:

  1. (a)

$a_{0}=1$,

  2. (b)

$a_{n}/a_{n+1}\leq 1$ for all $n\geq 0$,

  3. (c)

$\inf_{n\geq 0}a_{n}/a_{n+1}=0$, and

  4. (d)

$\lim_{n\to\infty}a_{n}^{1/n}=1$.

For instance, we can define $a_{n}:=k!$ whenever $k^{2}\leq n<(k+1)^{2}$. Then
properties (a) and (b) are obviously satisfied. Moreover, (c) holds since

$\frac{a_{(k+1)^{2}-1}}{a_{(k+1)^{2}}}=\frac{k!}{(k+1)!}=\frac{1}{k+1}\xrightarrow{k\to\infty}0.$

Finally, to see (d), notice that, if $k^{2}\leq n<(k+1)^{2}$, then

$1\leq a_{n}^{1/n}\leq(k!)^{1/k^{2}}\leq k^{1/k}\xrightarrow{k\to\infty}1.$

Next, we define a positive semi-definite kernel $K$ on ${\mathbb{D}}$ by

$K(z,w):=\sum_{n=0}^{\infty}a_{n}(z\overline{w})^{n}\quad(z,w\in{\mathbb{D}}).$

Condition (d) ensures that the power series defining $K$ has radius of
convergence $1$. Let ${\mathcal{H}}$ be the RKHS on ${\mathbb{D}}$ with
reproducing kernel $K$. It is well known and not hard to see that
${\mathcal{H}}$ consists of holomorphic functions on ${\mathbb{D}}$, and that
the monomials $(z^{n})_{n=0}^{\infty}$ form an orthogonal basis of
${\mathcal{H}}$ with

(4) $\|z^{n}\|^{2}=\frac{1}{a_{n}}\quad(n\geq 0),$

see e.g. [19, §4.4.2]. Since $a_{0}=1$, the reproducing kernel $K$ is
normalized at $0$.

We next show that the multiplier algebra ${\mathcal{M}}$ of ${\mathcal{H}}$
equals $H^{\infty}$. Certainly ${\mathcal{M}}$ is contractively contained in
$H^{\infty}$, as was already remarked in §2.2. On the other hand, condition
(b) above can be interpreted as saying that $a_{n}/a_{n+1}\leq b_{n}/b_{n+1}$
for all $n\geq 0$, where $b_{n}:=1$ for all $n\geq 0$, namely the power series
coefficients of the reproducing kernel of $H^{2}$. In this setting, [5,
Proposition 3.3] implies that $H^{\infty}$, the multiplier algebra of $H^{2}$,
is contractively contained in ${\mathcal{M}}$. Hence equality holds. In
particular, we see that ${\mathcal{M}}$ is dense in ${\mathcal{H}}$.

To construct the functional $\Lambda$, we first show that the (non-closed)
subspace $z{\mathcal{H}}$ has infinite codimension in ${\mathcal{H}}$. To this
end, notice that $M_{z}$ is an injective operator on ${\mathcal{H}}$, and
condition (c) and equation (4) together imply that it is not bounded below.
Therefore the range of $M_{z}$ is not closed and thus it has infinite
codimension in ${\mathcal{H}}$. Hence there exists a discontinuous linear
functional $\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ such that

$\Lambda(\alpha+zf)=\alpha\quad(\alpha\in{\mathbb{C}},~{}f\in{\mathcal{H}}).$

We finish the proof by showing the multiplicativity statement in (iii). To
this end, let $h\in{\mathcal{M}}=H^{\infty}$ and let $f\in{\mathcal{H}}$. Then
there exists $g\in H^{\infty}$ such that $h-h(0)=zg$. In particular,
$\Lambda(h)=h(0)$. Moreover, $hf=h(0)f+zgf$ and so

$\Lambda(hf)=\Lambda(h(0)f)+\Lambda(zgf)=h(0)\Lambda(f)=\Lambda(h)\Lambda(f),$

as claimed. ∎

##  6\. Point evaluations

Let ${\mathcal{H}}$ be a RKHS on $X$. If
$\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ is a point evaluation at a point
$x_{0}$ of $X$, namely $\Lambda(f):=f(x_{0})$ for all $f\in{\mathcal{H}}$,
then $\Lambda$ satisfies the hypothesis (i) of Theorem 1.2. Indeed, as
remarked earlier, if $f\in{\mathcal{H}}$ is cyclic, then $f(x)\neq 0$ for all
$x\in X$, and so $\Lambda(f)\neq 0$. Obviously $\Lambda(1)=1$ as well.

This raises the question of whether every linear functional $\Lambda$
satisfying the hypotheses of Theorem 1.2 is a point evaluation at some point
of $X$. The answer is affirmative in certain cases, notably the Hardy space
$H^{2}$ on the unit disk [17, Theorem 2.1] and, more generally, the
Drury–Arveson space $H^{2}_{d}$ on the unit ball in ${\mathbb{C}}^{d}$ for
$1\leq d\leq\infty$ (combine Theorem 1.2 above with [13, Lemma 5.3]). Under
the added assumption that $\Lambda$ is continuous, further results of this
kind have been established for various Banach function spaces on the unit disk
[17, Theorem 3.1] and on various domains in ${\mathbb{C}}^{n}$ [22, §5].

However, it is unrealistic to hope for an affirmative answer in the general
situation considered in Theorem 1.2. For example, if we think of $H^{2}$ as a
RKHS on the punctured disk ${\mathbb{D}}\setminus\\{x_{0}\\}$ for some fixed
point $x_{0}\in{\mathbb{D}}\setminus\\{0\\}$, then it still has a normalized
complete Pick kernel, and the linear functional $\Lambda(f):=f(x_{0})$ is non-
zero on cyclic functions, but is not given by point evaluation at any point of
${\mathbb{D}}\setminus\\{x_{0}\\}$.

A more ‘natural’ example is the case when
${\mathcal{H}}={\mathcal{D}}_{\zeta}$, the local Dirichlet space at a point
$\zeta\in{\mathbb{T}}$ [21]. This is a RKHS on the unit disk ${\mathbb{D}}$,
and by a result of Shimorin [23] it has a complete Pick kernel. In this case,
the functional $\Lambda(f):=\lim_{z\to\zeta}f(z)$ is well-defined and
continuous on ${\mathcal{D}}_{\zeta}$, and is non-zero on cyclic functions,
but it is not given by evaluation at any point of ${\mathbb{D}}$.

These examples suggest that, if we want to conclude that $\Lambda$ is a point
evaluation, then we need an assumption that ties $\Lambda$ more closely to the
set $X$. One possibility is to suppose that $\Lambda(f)\neq 0$ for every
$f\in{\mathcal{H}}$ that is _nowhere zero_ on $X$. This eliminates the two
counterexamples just mentioned. For $H^{2}$ on
${\mathbb{D}}\setminus\\{x_{0}\\}$ this is easy to see directly, and for
${\mathcal{D}}_{\zeta}$ it is a special case of [16, Theorem 1.3]. However, as
our final result shows, in the abstract setting of this paper, even this
stronger hypothesis is not enough the yield the desired conclusion.

######  Proposition 6.1.

There exist a RKHS ${\mathcal{H}}$ on a set $X$ and a linear functional
$\Lambda:{\mathcal{H}}\to{\mathbb{C}}$ with the following properties:

  1. (i)

The reproducing kernel $K:X\times X\to{\mathbb{C}}$ associated to
${\mathcal{H}}$ is a normalized complete Pick kernel.

  2. (ii)

The linear functional $\Lambda$ satisfies $\Lambda(1)=1$ and $\Lambda(f)\neq
0$ for each $f\in{\mathcal{H}}$ that is nowhere zero on $X$. Furthermore
$\Lambda$ is continuous on ${\mathcal{H}}$.

  3. (iii)

$\Lambda$ is not given by point evaluation at any point of $X$.

###### Proof.

Fix $d\geq 2$, and let $H^{2}_{d}$ denote the Drury–Arveson space on
${\mathbb{B}}_{d}$, the open unit ball in ${\mathbb{C}}^{d}$. Fix a point
$x_{0}\in{\mathbb{B}}_{d}\setminus\\{0\\}$, and set
$X:={\mathbb{B}}_{d}\setminus\\{x_{0}\\}$ and ${\mathcal{H}}:=H^{2}_{d}|_{X}$.

(i) The space ${\mathcal{H}}$ is a RKHS on $X$ whose kernel is normalized at
$0$ and has the complete Pick property.

(ii) Each function $f\in{\mathcal{H}}$ has a unique extension to a holomorphic
function on ${\mathbb{B}}_{d}$. Evaluating this extension at $x_{0}$ defines a
continuous linear functional $\Lambda$ on ${\mathcal{H}}$ such that
$\Lambda(1)=1$. Moreover, since holomorphic functions in two or more variables
have no isolated zeros, it follows that $\Lambda(f)\neq 0$ for every function
$f\in{\mathcal{H}}$ that does not vanish on $X$.

(iii) By considering its action on the coordinate functions, we see that
$\Lambda$ is not given by evaluation at any point of $X$. ∎

The counterexamples exhibited above are closely related to the notion of
_algebraic consistency_ or _maximal domains_. To elaborate somewhat, a non-
zero continuous linear functional $\Lambda$ on a RKHS ${\mathcal{H}}$ on $X$
is said to be _partially multiplicative_ if $\Lambda(fg)=\Lambda(f)\Lambda(g)$
for all $f,g\in{\mathcal{H}}$ satisfying $fg\in{\mathcal{H}}$. Clearly, point
evaluation at any point in $X$ defines a partially multiplicative functional.
The set $X$ is said to be a _maximal domain_ for ${\mathcal{H}}$ if every
partially multiplicative functional is of this type.

It was shown in [18, §2] that $X$ can always be enlarged to a maximal domain
by regarding ${\mathcal{H}}$ as a space of functions on its set of partially
multiplicative functionals. For concrete spaces of holomorphic functions on
domains in ${\mathbb{C}}^{d}$, one can sometimes determine the maximal domain
by considering all points $\lambda\in{\mathbb{C}}^{d}$ such that the
functional $p\mapsto p(\lambda)$, defined for all polynomials $p$, is
continuous with respect to the norm of ${\mathcal{H}}$; see [22, §3].
Moreover, in the case when ${\mathcal{H}}$ has a normalized complete Pick
kernel, there are alternative constructions of the maximal domain, using
weak-$*$ continuous characters on the multiplier algebra ${\mathcal{M}}$,
Gleason parts of the character space of ${\mathcal{M}}$, or a Zariski type
closure; see Proposition 4.2 and Remark 4.3 in [4]. For more information on
maximal domains, we refer the reader to [4], [9, Def.1.5], [13, §5], [18, §2],
and [22, §3].

## Acknowledgements

We thank the anonymous referees for their careful reading of the paper and for
several helpful suggestions of improvements.

Chu was supported by a CRM–Laval Postdoctoral Fellowship. Hartz was partially
supported by a GIF grant. Mashreghi was partially supported by an NSERC
Discovery Grant. Ransford was supported by grants from NSERC and the Canada
Research Chairs program.

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