# Some Modifications of Getzler’s Grading Technique

Andrés Larraín-Hubach111University of Dayton, OH 45469.
<EMAIL_ADDRESS>
###### Abstract

This paper reviews the grading technique developed by Getzler to prove the
local index theorem and shows how to adapt it to compute the leading terms of
asymptotic expansions of traces of heat kernels in two other situations.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Normal Coordinates and Synchronous frames
    2. 2.2 Geometric Dirac Operators
    3. 2.3 The Heat Kernel and Mehler’s Formula
    4. 2.4 Review of the Proof of the Local Index Theorem
  3. 3 Heat Kernels and Tensor Powers of Line Bundles
    1. 3.1 Statement of the Theorem
    2. 3.2 The Modified Grading and the Proof
  4. 4 Line Bundles and Heat Kernels on Odd-dimensional Manifolds
    1. 4.1 Definitions and Statement of the Theorem
    2. 4.2 Modification of the Grading and Proof

##  1 Introduction

The main objective of this work is to explain how the grading technique
developed by Ezra Getzler [3] to prove the local index theorem can be used to
study some problems of the following type: given a family of geometric Dirac
operators $\mathcal{D}_{s}$ that depends on a certain real parameter $s$, what
can be said about the leading order terms of the asymptotic expansion in
powers of $s$ of the heat kernel $e^{-\frac{t}{s}\mathcal{D}_{s}}(x,x)$
evaluated on the diagonal?

Understanding heat kernels, their traces and their values on the diagonal is
usually an important technical step that must be cleared before proving
results like the holomorphic Morse inequalities [4, Sec. 1.7], the asymptotics
of the spectral counting function [7, Prop. 4.2] and, more classically, the
local index theorem [1, Ch. 4]. Here we show how minor adaptations of the
grading technique can be used to compute the leading terms in all of these
expansions. The elementary proofs presented here are similar to each other and
display the grading technique as a “unifying theme”.

In section 2, we review classical results about geometric Dirac operators, the
asymptotic expansions of their heat kernels and the original definition of
Getzler’s grading technique. Finally, we review the well-known proof of the
local index theorem for a twisted Dirac operator over a spin manifold. Minor
modification of the ideas in this section will be used afterwards. There are
no new results in this section.

In section 3, we consider the following problem: given a complex manifold $M$
and a Hermitian bundle $\mathcal{E}=\Lambda^{*}T^{(0,1)*}M\otimes L^{\otimes
P}\otimes E$ with Clifford connection induced from the Chern connections on
$L$ and $E$, compute the leading order of the asymptotic expansion of the
diagonal values of $e^{-\frac{u}{p}(\mathcal{D}^{\mathcal{E}})^{2}}(x,x)$ as
$p\to\infty$, where $\mathcal{D}^{\mathcal{E}}$ is the associated geometric
Dirac operator. The answer, stated precisely in section 3.1, is [4, Thm.
1.6.1] and the argument presented there uses a method called Analytic
Localization which was originally developed by Bismut-Lebeau [2]. This
technique is powerful and can be used in several different contexts but it is
rather challenging to understand and relies on a considerable amount of
machinery from functional analysis. In the opinion of the author, the new
proof presented here of the same result is considerably easier since it only
uses the classical expansion of the heat kernel and a small modification of
the grading technique.

In section 4, we present a proof of [7, Thm. 4.1], explained in section 4.1.
Again, our argument is different from the one presented in that paper, in that
we modify the grading scheme and use the classical asymptotic expansion of the
heat kernel.

It is the expectation of the author that the reader will appreciate the power
of the grading technique as a useful and adaptable method to solve several
problems concerning heat kernels. For future work, it would be interesting to
use the grading technique to compute the leading terms of asymptotic
expansions of terms like
$\mathrm{Tr}(\mathcal{D}_{s}e^{-t\mathcal{D}_{s}^{2}})$ as $s$ goes to
infinity.

##  2 Preliminaries

In this section we fix notation and state the main properties and theorems to
be used in the following sections. We use Einstein’s summation whenever
indices are repeated. All manifolds are assumed compact, orientable and
without boundary. A good reference for this section is [6, Sec. 2.3].

###  2.1 Normal Coordinates and Synchronous frames

Let $(M,g)$ be a Riemannian manifold. The Levi-Civita connection on $TM$ is
denoted by $\nabla$. Using normal coordinates $\\{\mathrm{x}_{l}\\}_{l=1}^{n}$
around $y$, we can identify $y$ with $0\in\mathbb{R}^{n}.$ This will be done
implicitly in what follows. The frame
$\\{\frac{\partial}{\partial\mathrm{x}_{l}}\\}_{l=1}^{n}$ is orthonormal at
$y$. In fact, in these coordinates
$g_{ij}(\mathrm{x})=g(\frac{\partial}{\partial\mathrm{x}_{i}},\frac{\partial}{\partial\mathrm{x}_{j}})(\mathrm{x})=\delta_{ij}+\mathcal{O}(\mathrm{x}^{2})$
and the symbols of the Levi-Civita connection symbols satisfy
$\Gamma_{ij}^{k}=\mathcal{O}(\mathrm{x}).$

The local radial function is defined by
$r^{2}(\mathrm{x})=\sum_{j=1}^{n}\mathrm{x}_{j}^{2}.$ It is a consequence of
Gauss lemma that $r(\mathrm{x})=d(x,y)$ where $d$ is the distance function
induced by the metric $g$. The radial vector field is defined by
$r\partial_{r}=\mathrm{x}_{l}\frac{\partial}{\partial\mathrm{x}_{l}}.$ Taking
the $\\{\frac{\partial}{\partial\mathrm{x}_{l}}\\}_{l=1}^{n}$ at $y$ and using
parallel translation along geodesics that emanate from $y$ results in a local
orthonormal frame that we denote by $\\{e_{l}\\}_{l=1}^{n}$ with dual
orthonormal coframe denoted by $\\{\omega^{l}\\}_{l=1}^{n}.$

Let $E\to M$ be a Hermitian vector bundle with unitary connection
$\nabla^{E}=d+A$. Take an orthonormal frame of the fiber of $E$ over $y\in M$
and extend it to a local frame using again parallel translations along
geodesics emanating from $y$. In this local frame, called synchronous, the
connection symbols satisfy

$\nabla^{E}_{i}=\frac{\partial}{\partial\mathrm{x}_{i}}-\frac{1}{2}\mathrm{x}^{j}F(\frac{\partial}{\partial\mathrm{x}_{i}},\frac{\partial}{\partial\mathrm{x}_{j}})+\mathcal{O}(\mathrm{x}^{2})=\frac{\partial}{\partial\mathrm{x}_{i}}-\frac{1}{2}\mathrm{x}_{j}F_{ij}+\mathcal{O}(\mathrm{x}^{2}),$
(1)

where $F$ denotes the curvature two-form of $\nabla^{E}.$ (See [5, Prop.
12.22]).

###  2.2 Geometric Dirac Operators

In this section, $(M,g)$ is a closed Riemannian manifold of
$\dim_{\mathbb{R}}M=n=2m.$ We denote by $CL(TM)\otimes\mathbb{C}$ the
complexified Clifford algebra bundle of $TM.$ Our conventions for Clifford
algebras are the same as those of [5]. Additionally, we have a Hermitian
bundle $\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}\to M$ with metric
$h^{\mathcal{E}}$ and compatible connection $\nabla^{\mathcal{E}}$ satisfying
the following properties:

  * •

The bundle $\mathcal{E}$ is a Clifford bundle. In other words, there is a
homomorphism of complex algebras
$c:CL(TM)\otimes\mathbb{C}\to\mathrm{End}\,\mathcal{E}$ called Clifford
multiplication. The Clifford multiplication by a tangent vector field $X$ is
denoted by $c(X)\in\Gamma(\mathrm{End}\,\mathcal{E})$. One also assumes that
$c(X)$ is skew-symmetric with respect to $h^{\mathcal{E}}$ and
$c(X):\mathcal{E}^{\pm}\to\mathcal{E}^{\mp}.$

  * •

The connection $\nabla^{\mathcal{E}}$ preserves the splitting
$\mathcal{E}=\mathcal{E}^{+}\oplus\mathcal{E}^{-}.$

  * •

The connection $\nabla^{\mathcal{E}}$ is a Clifford connection. In other
words,
$\nabla^{\mathcal{E}}_{X}(c(Y)\psi)=c(\nabla_{X}Y)\psi+c(Y)\nabla^{\mathcal{E}}_{X}\psi$,
where $X,Y$ are vector fields on $M$ and $\psi\in\Gamma(\mathcal{E}).$

On $y\in M$, we have that $\mathcal{E}_{y}\cong S\otimes W_{y}$, where $S$ is
the spinor space and $W$ is the so-called twisting bundle. The $S$ glue to a
globally defined vector bundle when $M$ is spin. Furthermore, the following
isomorphism of complex algebras holds [5, Ch. 4]

$\mathrm{End}(\mathcal{E}_{y})\cong(CL(T_{y}M)\otimes\mathbb{C})\otimes\mathrm{End}(W_{y}).$
(2)

The algebra $CL(T_{y}M)\otimes\mathbb{C}$ is generated by $c(e_{j})=c^{j}$ for
$j=1\ldots,n.$ The $\\{e_{j}\\}$ is the local orthonormal frame defined in
section 2.1. Using normal coordinates and a synchronous frame on $\mathcal{E}$
we get the following local expression for the Clifford connection
$\nabla^{\mathcal{E}}$:

$\nabla^{\mathcal{E}}_{\frac{\partial}{\partial\mathrm{x}_{i}}}=\frac{\partial}{\partial\mathrm{x}_{i}}+\frac{1}{4}\Gamma_{ij}^{k}(\mathrm{x})c^{j}c^{k}+h_{i}(\mathrm{x}),$
(3)

where $h_{i}(\mathrm{x})=\mathcal{O}(\mathrm{x}).$ See [1, Lemma 4.14].

With this information, we define the Dirac operator on the Clifford bundle
$\mathcal{E}$ by

$\mathcal{D}^{\mathcal{E}}=\sum_{i=1}^{n}c(e_{i})\nabla^{\mathcal{E}}_{e_{i}}$
(4)

This produces an elliptic, Fredholm and self-adjoint operator. Let
$\mathcal{D}^{\pm}=\mathcal{D}^{\mathcal{E}}|_{\mathcal{E}^{\pm}}$. We record
here the following crucial identity known as Lichnerowicz formula

$(\mathcal{D}^{\mathcal{E}})^{2}=(\nabla^{\mathcal{E}})^{*}\nabla^{\mathcal{E}}+c(F^{\mathcal{E}}),$
(5)

where $F^{\mathcal{E}}$ denotes the curvature of the connection
$\nabla^{\mathcal{E}}$. The notation $c(F^{\mathcal{E}})$ means that, if
$F^{\mathcal{E}}=\frac{1}{2}F(e_{i},e_{j})\,\omega^{i}\wedge\omega^{j}$ then
$c(F^{\mathcal{E}})=\frac{1}{2}F(e_{i},e_{j})c(e_{i})c(e_{j}).$ Here we
identify $TM$ and $T^{*}M$ using the metric $g$.

###  2.3 The Heat Kernel and Mehler’s Formula

The proofs of the results in this section can be found in [6, Ch. 3] and [5,
Ch. 7]. The operator $\mathcal{D}^{\mathcal{E}}$ is Fredholm and self-adjoint
with discrete spectrum. Using the functional calculus one can define the
smoothing operator $e^{-t(\mathcal{D}^{\mathcal{E}})^{2}}$ for every $t>0$.
This operator has a smooth integral kernel denoted by
$k_{t}(x,y)\in\mathrm{End}(\mathcal{E}_{y},\mathcal{E}_{x})$ for every
$(x,y)\in M\times M$ and $t>0$, such that for $\psi\in\Gamma(\mathcal{E})$ one
has

$(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}}\psi)(x)=\int_{M}k_{t}(x,y)\psi(y)\mathrm{vol}_{y},$

where $\mathrm{vol}_{y}$ denotes the volume form in the $y$-variable induced
from the metric $g.$ The following approximation of the heat kernel on
$\mathbb{R}^{n}$ is going to be used frequently

$q_{t}(x,y)=\frac{1}{(4\pi t)^{n/2}}\exp\\{\frac{-d(x,y)^{2}}{4t}\\},$ (6)

where $d(x,y)$ denotes the distance induced by the metric between the points
$x$ and $y$ on the manifold. We use $inj$ to denote the injectivity radius of
$M$.

The following theorem lists several important properties of the heat kernel
$k_{t}(x,y).$

######  Theorem 1.

The kernel $k_{t}(x,y)$ is differentiable in $t$ and smooth in the variables
$x,y$. It also satisfies the following:

  1. 1.

If $(\mathcal{D}^{\mathcal{E}})^{2}_{x}$ denotes the operator acting on the
$x$-variable, then

$(\partial_{t}+(\mathcal{D}^{\mathcal{E}})_{x}^{2})k_{t}(x,y)=0.$

  2. 2.

For every $\psi\in\Gamma(\mathcal{E})$ one has

$\lim_{t\to 0}k_{t}(x,y)\psi(y)\mathrm{vol}_{y}\to\psi(x),$

uniformly in $x.$

  3. 3.

The operator $e^{-t(\mathcal{D}^{\mathcal{E}})^{2}}$ is trace class and its
operator trace is given by

$\mathrm{Tr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})=\int_{M}\mathrm{tr}_{\mathcal{E}}\,k_{t}(y,y)\mathrm{vol}_{y},$
(7)

where $\mathrm{tr}_{\mathcal{E}}$ denotes the usual trace in
$\mathrm{End}(\mathcal{E}_{y}).$

  4. 4.

There are sections
$\Theta_{j}(x,y)\in\mathrm{End}(\mathcal{E}_{y},\mathcal{E}_{x})$ for
$j=0,1,2,\ldots$ defined in a neighborhood of the diagonal in $M\times M$ such
that for every $N>m=n/2$, there is a constant $C_{N}$ such that

$\|\partial^{k}_{t}(k_{t}(x,y)-\chi(x,y)q_{t}(x,y)\sum_{j=0}^{N}t^{j}\Theta_{j}(x,y))\|_{C^{l}}\leq
C_{N}t^{N-m-l/2-k}.$

Here, $\chi(x,y)=\chi(d(x,y))$ where $\chi(r)=1$ for $r\leq\frac{inj^{2}}{4}$
and $\chi(r)=0$ for $r\geq inj^{2}.$ Also, $\|\cdot\|_{C^{l}}$ denotes the
$C^{l}$ norm on $M\times M$. In other words, there is an asymptotic expansion

$k_{t}(x,y)\sim\chi(x,y)q_{t}(x,y)\sum_{j=0}^{\infty}t^{j}\Theta_{j}(x,y)$ (8)

  5. 5.

Using normal coordinates $\\{\mathrm{x}_{l}\\}_{l=1}^{n}$ centered at $y\in M$
and simplifying notation by using $\Theta_{j}(\mathrm{x})$ instead of
$\Theta_{j}(x,y)$, one has the following recursion

$\nabla_{\partial_{r}}(r^{j}|g|^{1/4}\Theta_{j})=\begin{cases}0,&j=0\\\
-r^{j-1}|g|^{1/4}(\mathcal{D}^{\mathcal{E}})^{2}\Theta_{j-1},&j\geq
1,\end{cases}$

where $|g|=\mathrm{Det}(g)$. These are ordinary differential equations along
geodesics emanating from $y$. The boundary condition
$\Theta_{0}(y,y)=\Theta_{0}(0)=\mathrm{id_{\mathcal{E}}}$ implies that the
$\Theta_{j}$ are uniquely defined for all $j$. Also, the smoothness of the
heat kernel implies that $\Theta_{0}(\mathrm{x})=C|g|^{-1/4}(\mathrm{x}).$ For
the other values of $j$, one can integrate along the geodesic joining $0$ and
$\mathrm{x}$ to obtain

$\Theta_{j}(\mathrm{x})=-\frac{1}{|g|^{1/4}}\int_{0}^{1}s^{j-1}|g|^{1/4}(s\mathrm{x})((\mathcal{D}^{\mathcal{E}})^{2}\Theta_{j-1})(s\mathrm{x})\,ds$
(9)

  6. 6.

The famous Mckean-Singer theorem states that

$\mathrm{Ind}(\mathcal{D}^{+})=\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})=\int_{M}\mathrm{str}\,k_{t}(y,y)\,\mathrm{vol}_{y},$
(10)

where the super trace of a map $T\in\mathrm{End}(\mathcal{E}_{y})$ such that
$T|_{\mathcal{E}_{y}^{\pm}}\subset\mathcal{E}_{y}^{\pm}$ is defined by
$\mathrm{str}(T)=\mathrm{tr}(T|_{\mathcal{E}_{y}^{+}})-\mathrm{tr}(T|_{\mathcal{E}_{y}^{-}})$.
In particular, the supertrace of the heat kernel is constant in $t$.

Finally, we record Mehler’s formula which will be used later [5, Ch. 12]. Let
$R$ be an $n\times n$ antisymmetric matrix with coefficients in a commutative
algebra $\mathcal{A}$ and $F$ be an $N\times N$ matrix with coefficients also
in $\mathcal{A}$. Consider the operator

$H=-(\partial_{i}+\frac{1}{4}\mathrm{x}_{j}R_{ij})^{2}+F$

acting on functions
$\mathbb{R}^{n}\to\mathcal{A}\otimes\mathrm{End}(\mathbb{C}^{N}).$ Then the
heat kernel of $H$, denoted by $p_{t}(\mathrm{x})$ has the following form

$p_{t}(\mathrm{x})=(4\pi
t)^{-n/2}\sqrt{\det\big{(}\frac{tR/2}{\sinh(tR/2)}\big{)}}\,e^{-\frac{1}{4t}\langle\mathrm{x}|\frac{tR}{2}\coth\frac{tR}{2}|\mathrm{x}\rangle}e^{-tF}.$
(11)

###  2.4 Review of the Proof of the Local Index Theorem

Let $(M,g)$ be a spin manifold with a fixed spin structure. Let $S=S^{+}\oplus
S^{-}\to M$ denote the spinor bundle with connection $\nabla^{S}$ induced by
the Levi-Civita connection. Additionally, take $(E,h^{E},\nabla^{E})$ be a
Hermitian bundle with unitary connection. Following the first section, we work
with the bundle

$\mathcal{E}=S\otimes E=(S^{+}\otimes E)\oplus(S^{-}\otimes E).$

In other words, $\mathcal{E}^{\pm}=S^{\pm}\otimes E.$ The tensor product
connection induces a Clifford connection $\nabla^{\mathcal{E}}.$ Locally, we
have [1, Sect. 3.2 and 3.3]

$\nabla^{\mathcal{E}}_{\frac{\partial}{\partial\mathrm{x}_{i}}}=\frac{\partial}{\partial\mathrm{x}_{i}}+\frac{1}{4}\Gamma_{ij}^{k}(\mathrm{x})c^{j}c^{k}+\Gamma^{E}_{i}$

The Dirac operator $\mathcal{D}^{\mathcal{E}}$ is defined as in (4), and (5)
becomes

$(\mathcal{D}^{\mathcal{E}})^{2}=(\nabla^{\mathcal{E}})^{*}\nabla^{\mathcal{E}}+\frac{1}{4}\mathrm{Scal}+c(F^{E}),$
(12)

where $\mathrm{Scal}$ denotes the scalar curvature of $M.$

Before we continue, we review some standard notation: let
$I=(i_{1},\ldots,i_{k})$ be a multi-index with $1\leq i_{1}\leq\ldots\leq
i_{k}\leq n$, then
$\mathrm{x}^{I}=\mathrm{x}_{i_{1}}\mathrm{x}_{i_{2}}\ldots\mathrm{x}_{i_{k}}$,
$\partial_{\mathrm{x}}^{I}=\frac{\partial}{\partial\mathrm{x}_{i_{1}}}\ldots\frac{\partial}{\partial\mathrm{x}_{i_{k}}}$
and $c^{I}=c^{i_{1}}\ldots c^{i_{k}}.$ Capital letters as exponents will
denote multi-indices from now on. Notice that in $c^{I}$ we also assume $1\leq
i_{1}<\ldots<i_{k}\leq n$ since otherwise some of the $c^{i}$ would simplify
using the Clifford algebra rules. The number $|I|=k$ in $c^{I}$ is the number
of Clifford factors.

The objective is to compute
$\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})=\int_{M}\mathrm{str}\,k_{t}(y,y)\,\mathrm{vol}_{y}$
which gives the index of $\mathcal{D}^{+}$ from Mckean-Singer’s theorem.

It is a well-known fact [5, Lemma 11.5] that $\mathrm{str}(c^{I})=0$ if
$|I|\neq n$ and $\mathrm{str}(c^{1}\ldots c^{n})=(-2i)^{n/2}$. For this
reason, one needs to identify only those terms in the expansion (8) that have
the maximum possible number of Clifford factors and which do not vanish when
evaluated at $\mathrm{x}=0$, which corresponds to the point $y\in M$ at the
center of the normal coordinate system. In other words, for some multi-index
$I$, a summand of the form

$a\mathrm{x}^{I}c^{1}\ldots c^{n},$

with $a\in\mathrm{End}(E_{y})$ only contributes to
$\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})$ if $|I|=k=0$ because
$\mathrm{sTr}$ also includes an evaluation at $\mathrm{x}=0.$

This observation is the motivation of the following idea of Ezra Getzler aimed
at extracting the terms mentioned above [3].

######  Definition 2.

The $cG$-grading is defined as follows

$\displaystyle\mathrm{ord}_{cG}(\mathrm{x}_{i})=-1$
$\displaystyle\mathrm{ord}_{cG}(\frac{\partial}{\partial\mathrm{x}_{i}})=1$
$\displaystyle\mathrm{ord}_{cG}(c^{i})=1$ (13)

and coefficients in $\mathrm{End}(E_{y})$ have $\mathrm{ord}_{cG}=0.$ For
example, $\mathrm{ord}_{cG}(x^{I}c^{J}\partial_{\mathrm{x}}^{K})=|J|-|I|+|K|$
and the G-order of a sum is the maximum of the G-orders of the summands.

######  Theorem 3.

Given two differential operators $A,B$ written formally as

$\displaystyle A$
$\displaystyle=\sum_{I,J,K}a_{I,J,K}\mathrm{x}^{I}c^{J}\partial_{\mathrm{x}}^{K}$
$\displaystyle B$
$\displaystyle=\sum_{S,T,U}b_{S,T,U}\mathrm{x}^{S}c^{T}\partial_{\mathrm{x}}^{U},$

with $a_{l,I,J}$ and $b_{k,S,T}$ in $\mathrm{End}(E_{y})$ independent of
$\mathrm{x}$ and without Clifford factors. then

$\mathrm{ord}_{cG}(A\circ B)\leq\mathrm{ord}_{cG}(A)+\mathrm{ord}_{cG}(B).$

###### Proof.

The Clifford factors are constant in a synchronous frame [1, Lemma 4.13].
Consider first the composition of two of the summands, that is

$(a_{I,J,K}x^{I}c^{J}\partial_{\mathrm{x}}^{K})\circ(b_{S,T,U}x^{S}c^{T}\partial_{\mathrm{x}}^{U})=a_{I,J,K}b_{S,T,U}x^{I+S}c^{J+T}\partial_{K+U}+\ldots$

where the dots mean summands with fewer derivatives. The sum of multi-indices
is defined in the obvious way. In these terms, some derivatives were used
differentiating the $x^{S}$. Each differentiation lowers the
$\mathrm{x}$-exponent by one or cancels the whole term. For example,
$\partial_{i}(x^{P})=0$ if $i\notin P$. Therefore, either
$\mathrm{ord}_{cG}$-order remains the same (losses one derivative and one
factor of $x$) or the term vanishes. Also notice that the number of Clifford
factors in $c^{J+T}$ is less than or equal to $|J|+|T|$. This implies the
result.

∎

This definition implies that $\mathrm{ord}_{cG}(\nabla^{\mathcal{E}}_{i})\leq
1$, $\mathrm{ord}_{cG}(\mathcal{D}^{\mathcal{E}})\leq 2$ and, using formula
(12), $\mathrm{ord}_{cG}((\mathcal{D}^{\mathcal{E}})^{2})\leq 2.$

Since $|g|(y)\neq 0$, we have that $\mathrm{ord}_{cG}(\Theta_{0})\leq 0.$
Using induction, it follows from (9) that $\mathrm{ord}_{cG}(\Theta_{j})\leq
2j.$

From the Mckean-Singer theorem it follows that
$\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})$ is independent of $t$.
Using (8) one gets

$\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})=(4\pi)^{-n/2}\int_{M}\mathrm{str}\,\Theta_{n/2}(y,y)\,\mathrm{vol}_{y}$
(14)

A term in $\Theta_{n/2}(y,y)$ with a chance of contributing to the integral on
the right hand side of (14), after taking super-trace and evaluating at
$\mathrm{x}=0$ (which is the same as evaluating at $y$), must be of the form
$ac^{1}\ldots c^{n}$ with $0\neq a\in\mathrm{End}(E_{y})$. In this case,
$\mathrm{ord}_{cG}(ac^{1}\ldots c^{n})=n$. Therefore, the terms with non-
vanishing sTr are among those with $\mathrm{ord}_{cG}=n$. Theorem 3 implies
that the terms of $\mathrm{ord}_{cG}=n$ in $\theta_{n/2}$ are obtained from
taking the terms of maximum $\mathrm{ord}_{cG}$ in each $\Theta_{j}$ and
following the recurrence (9) restricted only to those terms.

Basically, this is achieved by starting with the same $\Theta_{0}$ and
following (9) with $(\mathcal{D}^{\mathcal{E}})^{2}$ replaced by its terms of
$\mathrm{ord}_{cG}=2$. That is

$H_{1}=-(\frac{\partial}{\partial_{\mathrm{x}_{i}}}+\frac{1}{4}\Gamma_{ij}^{k}c^{j}c^{k})^{2}+c(F^{E})$
(15)

This can be simplified even further by noticing that a commutator
$[c^{i}c^{j},c^{k}c^{l}]$ can have either two Clifford factors or vanish
identically. In both cases, the $\mathrm{ord}_{cG}$ of the commutator is not
the expected $4$ but instead is either $0$ or $2$. Since commutators lower
$\mathrm{ord}_{cG}$ and we only want the terms with the highest possible
$\mathrm{ord}_{cG}$, we can replace the Clifford algebra by the exterior
algebra instead. This means that we need to analyze the heat kernel of the
operator

$H=-(\frac{\partial}{\partial_{\mathrm{x}_{i}}}+\frac{1}{4}\mathrm{x}_{j}R_{ij})^{2}+F^{E}$
(16)

Finally, Mehler’s formula and (16) imply that

$\mathrm{ind}(\mathcal{D}^{+})=\mathrm{sTr}(e^{-t(\mathcal{D}^{\mathcal{E}})^{2}})=(4\pi)^{-n/2}(-2i)^{n/2}\int_{M}\sqrt{\det\big{(}\frac{R/2}{e^{R/2}-e^{-R/2}}\big{)}}\,e^{-F}\,\mathrm{vol},$
(17)

from which the index theorem follows. See [5, pp. 162].

##  3 Heat Kernels and Tensor Powers of Line Bundles

In this section we give a new and simple proof of Theorem 1.6.1 in [4]. Our
argument only uses the expansion (8) and a small modification of the
$cG$-grading explained in the previous section.

###  3.1 Statement of the Theorem

Let $(M,g)$ be a complex manifold whose almost complex structure is compatible
with the Riemannian metric. Let $(L,h^{L})$ be a holomorphic line bundle over
$M$ with Chern connection $\nabla^{L}$, and $(E,h^{E})$ be holomorphic vector
bundle over $M$ with Chern connection $\nabla^{E}.$ Let $T^{*(0,1)}M$ be the
$(0,1)$-part of the cotangent bundle of $M$ with the induced connection coming
from the Levi-Civita connection. Similarly, one can induce a connection on the
wedge powers $\Lambda^{*}T^{*(0,1)}M.$ Given $y\in M$, we use these
connections to generate synchronous local orthonormal frames on each of these
bundles.

In this setup, the bundle to be considered is

$\mathcal{E}_{p}=\Lambda T^{*(0,1)}M\otimes L^{\otimes p}\otimes E,$ (18)

where $L^{\otimes p}$ denotes the $p^{th}$ tensor power of $L$ with itself.
The bundle $\mathcal{E}_{p}$ is a Clifford bundle with Clifford multiplication
given by

$c(e_{j})=\sqrt{2}\epsilon(\omega^{(0,1)j})-\sqrt{2}\iota_{e_{j}^{(0,1)}},$

where $\epsilon$ and $\iota$ denote exterior and interior product respectively
and $\omega^{(0,1)j}$, $e_{j}^{(0,1)}$ denote the $(0,1)$ parts of the
covector and the vector respectively. Also, we use the tensor product of the
connections on $\Lambda^{*}T^{*(0,1)}M$, $L$ and $E$ to generate a Clifford
connection $\nabla^{\mathcal{E}_{p}}$ on $\mathcal{E}_{p}.$

The associated geometric Dirac operator to $\nabla^{\mathcal{E}_{p}}$ and the
Clifford product above is defined as in (4) and will be denoted by
$\mathcal{D}_{p}$ to simplify notation.

Let $F^{L}$ denote the curvature of $\nabla^{L}$, then the endomorphism
$\dot{F}^{L}$ is defined as follows: take $X,Y$ vector fields in $T^{(1,0)}M$
then

$F^{L}(X,\bar{Y})=g(\dot{F}^{L}X,\bar{Y}).$

Notice that $\dot{F}^{L}\in\mathrm{End}({T^{(1,0)}M}).$ Set
$\tau(y)=\mathrm{tr}(\dot{F}^{L}|_{y})$ for any $y\in M.$ Also, if
$\\{Z_{j}\\}_{j=1}^{m}$ is any orthonormal frame of $T^{(1,0)}M$ then one
defines

$\omega_{d}=-F^{L}(Z_{l},\bar{Z}_{m})\epsilon(\bar{Z}^{l})\iota_{\bar{Z}_{m}},$

where $\bar{Z}^{l}\in T^{(0,1)*}M$ is the dual covector. With all this
notation, we can state the main theorem

######  Theorem 4.

[4, thm 1.6.1] For each $u>0$ and all $k\in\mathbb{N}$ we have as $p\to\infty$
that

$e^{-\frac{u}{p}\mathcal{D}_{p}^{2}}(y,y)=k_{u/p}(y,y)=(2\pi)^{-n/2}p^{n/2}\frac{e^{2u\omega_{d}}\det(\dot{F}^{L})}{\det{(1-e^{-2u\dot{F}^{L}}})}\otimes\mathrm{id}_{E}+o(p^{n/2})$
(19)

This asymptotic expansion is a key step in the proof of the holomorphic Morse
inequalities explained in [4, section 1.7]. The apparent discrepancy in the
exponents of $p$ occurs because we denote by $n$ the real dimension of $M$
while the authors in [4] use $n$ to denote the complex dimension.

###  3.2 The Modified Grading and the Proof

Now we explain the modification of the grading scheme that gives another proof
of theorem 4. First of all, notice that (19) does not involve traces or super
traces. For this reason, counting the number of Clifford factors of the form
$c^{j}$ does not give any relevant information. Instead, what we really care
about is counting the exponents of $p$.

######  Definition 5.

The $p$-grading is defined as follows

$\displaystyle\mathrm{ord}_{pG}(\mathrm{x}_{j})=-1$
$\displaystyle\mathrm{ord}_{pG}(\frac{\partial}{\partial\mathrm{x}_{l}})=1$
$\displaystyle\mathrm{ord}_{pG}(p)=2$ (20)

and coefficients in $\mathrm{End}(\Lambda T_{y}^{*(0,1)}M\otimes L_{y}\otimes
E_{y})$ have $\mathrm{ord}_{pG}=0.$ The $p$-grading of a sum is the maximum of
the $p$-gradings of the summands.

The proof of the following result is almost identical to that of theorem 3.

######  Theorem 6.

Given two differential operators $A,B$ written formally as

$\displaystyle A$
$\displaystyle=\sum_{l,I,J}a_{l,I,J}p^{l}\mathrm{x}^{I}\partial_{\mathrm{x}}^{J}$
$\displaystyle B$
$\displaystyle=\sum_{k,S,T}b_{k,S,T}p^{k}\mathrm{x}^{S}\partial_{\mathrm{x}}^{T},$

with $a_{l,I,J}$ and $b_{k,S,T}$ in $\mathrm{End}(\Lambda
T_{y}^{*(0,1)}M\otimes L_{y}\otimes E_{y})$ independent of $p.$ then

$\mathrm{ord}_{pG}(A\circ B)\leq\mathrm{ord}_{pG}(A)+\mathrm{ord}_{pG}(B).$

Locally, the connection $\nabla^{\mathcal{E}_{p}}$ can be written as

$\nabla^{\mathcal{E}_{p}}_{\frac{\partial}{\partial\mathrm{x}_{i}}}=\frac{\partial}{\partial\mathrm{x}_{i}}+\frac{1}{4}\Gamma_{ij}^{k}c^{j}c^{k}+\frac{1}{2}\Gamma^{\det}_{i}+p\Gamma^{L}_{i}+\Gamma^{E}_{i},$
(21)

where $\Gamma^{\det}$ is the connection one-form on the determinant line
bundle $\det(T^{(1,0)}M).$ Notice that, with respect to the decomposition (3),
we have
$h_{i}(\mathrm{x})=\frac{1}{2}\Gamma^{\det}_{i}+p\Gamma^{L}_{i}+\Gamma^{E}_{i}$
and $\mathrm{ord}_{pG}(\frac{1}{2}\Gamma^{\det}_{i}+\Gamma^{E}_{i})\leq-1.$
This implies that

$\mathrm{ord}_{pG}(\nabla^{\mathcal{E}_{p}}_{\frac{\partial}{\partial\mathrm{x}_{i}}})\leq
1,$

and its highest order summands are
$\frac{\partial}{\partial\mathrm{x}_{i}}+p\Gamma^{L}_{i}.$ Theorem 6 implies
then that $\mathrm{ord}_{pG}(\mathcal{D}^{2}_{p})\leq 2$ and the recurrence
(9) implies that $\mathrm{ord}_{pG}(\Theta_{j})\leq 2j.$ The summands in
$\Theta_{j}$ of $\mathrm{ord}_{pG}=2j$ that do not vanish on the diagonal will
have $j$ as the highest possible exponent of $p.$

From the expansion (8) we get

$k_{u/p}(y,y)\sim\frac{p^{n/2}}{(4\pi
u)^{n/2}}\sum_{j=0}^{\infty}\frac{u^{j}}{p^{j}}\Theta_{j}(y,y),$ (22)

which implies that the highest possible exponent of $p$ in (22) is $n/2.$ This
exponent can be reached only using the terms of highest $\mathrm{ord}_{pG}$ in
each $\Theta_{j}$.

In other words, we need to investigate the heat kernel of the operator

$-(\frac{\partial}{\partial\mathrm{x}_{i}}+p\Gamma^{L}_{i})^{2}+pc(F^{L})=-(\frac{\partial}{\partial\mathrm{x}_{i}}+p\Gamma^{L}_{i})^{2}-p(2\omega_{d}+\tau)$

(See [4, pp. 46]). Using (1) and dropping terms of lower $\mathrm{ord}_{pG}$
clarifies that we need to understand the heat kernel of 222We are omiting
$\otimes\mathrm{id}_{E}$ in this notation.

$Q_{p}=-(\frac{\partial}{\partial\mathrm{x}_{i}}-\frac{p}{2}\mathrm{x}_{j}F^{L}_{ij})^{2}-p(2\omega_{d}+\tau)$
(23)

Using Mehler’s formula again, the integral kernel of $e^{-tQ_{p}^{2}}$ is
given by

$e^{-tQ_{p}^{2}}(y,y)=\mathrm{id_{E}}\otimes(2\pi
t)^{-n/2}\sqrt{\det\big{(}\frac{tpF^{L}}{e^{tpF^{L}}-e^{-tpF^{L}}}\big{)}}\,e^{tp(2\omega_{d}(y)+\tau(y))}$

We have that $\det(F^{L})=\det^{2}(\dot{F}^{L})$ so we get

$e^{-tQ_{p}^{2}}(y,y)=\mathrm{id_{E}}\otimes(2\pi
t)^{-n/2}{\det\big{(}\frac{tp\dot{F}^{L}}{1-e^{-2tp\dot{F}^{L}}}\big{)}}\det(e^{-tp\dot{F}^{L}})\,e^{tp(2\omega_{d}(y)+\tau(y))}$

Also, $\det(e^{tp\dot{F}^{L}})|_{y}=e^{\tau(y)}$ and the corresponding terms
cancel. Therefore,

$e^{-tQ_{p}^{2}}(y,y)=\mathrm{id_{E}}\otimes(2\pi
t)^{-n/2}{\det\big{(}\frac{tp\dot{F}^{L}}{1-e^{-2tp\dot{F}^{L}}}\big{)}}\,e^{2tp\omega_{d}(y)}$

Finally, replace $t$ by $u/p$ and notice that
$\det{u\dot{F}^{L}}=u^{n/2}\det{\dot{F}^{L}}$. This gives the leading order
term in (19) and proves theorem 4.

##  4 Line Bundles and Heat Kernels on Odd-dimensional Manifolds

In this section we show one more application of the grading technique by
giving an alternative proof of [7, Formula (4.1)]. One major difference in
this case is that the base manifold $Y$ is odd-dimensional. Again, a small
modification of the definition of $\mathrm{ord}_{cG}$ will give the leading
order term of the asymptotic expansion of a heat kernel in powers of a certain
parameter.

###  4.1 Definitions and Statement of the Theorem

Let $(Y,g)$ be a Riemannian spin manifold with $\dim_{\mathbb{R}}=n=2m+1.$ One
fixes the spin structure on $Y$ and denotes by $S\to Y$ the associated spinor
bundle with induced connection $\nabla^{S}$. Let $L\to Y$ be a Hermitian line
bundle with a unitary connection $\nabla_{0}=d+A_{0}$. Let $a$ be a
$i\mathbb{R}$-valued one-form on $Y$. Define a family of connections on
$S\otimes L$ by

$\nabla^{r}=\nabla^{S}\otimes\mathrm{Id}_{L}+\mathrm{Id}_{S}\otimes(\nabla^{0}+ra),$

where $r$ is a real parameter. Using the family $\nabla^{r}$, we define a
family of Dirac operators by

$\mathcal{D}^{r}=c^{k}\,\nabla^{r}_{e_{k}}$ (24)

as in (4). Notice that the bundle $\mathcal{E}=S\otimes L$ in this case is not
$\mathbb{Z}_{2}$-graded.

The heat kernel $e^{-t(\mathcal{D}^{r})^{2}}$ is defined using functional
calculus as before and all the statements in theorem 1 remain valid with the
exception of the McKean-Singer formula. Also, Mehler’s formula remains valid
with $n$ odd. In the next section, we present a new proof of the following
result

######  Theorem 7.

[7, theorem 4.1] For any fixed $t>0$, one has

$\lim_{r\to\infty}r^{-n/2}\mathrm{Tr}(e^{-\frac{t}{r}(\mathcal{D}^{r})^{2}})=(4\pi
t)^{-n/2}\int_{Y}\sqrt{\det\dfrac{tA_{y}}{\tanh tA_{y}}}\,\mathrm{vol}_{y}$
(25)

Here $A_{y}\in\mathrm{End}(T_{y}Y)$ is defined by
$ida(X_{1},X_{2})(y)=g(X_{1},AX_{2})$ for all $X_{1},X_{2}\in T_{y}Y.$ The
convergence is uniform on compact intervals of $t\in(0,\infty).$

###  4.2 Modification of the Grading and Proof

Fix $y\in Y$ and use this point as the center of a normal coordinate system.
Around $y$, define a synchronous orthonormal frame on $L$ by using
$\nabla^{0}$ and parallel transport along geodesics emanating from $y$. We use
the same notation $\\{\mathrm{x}_{l}\\}_{l=1}^{n}$ as before to denote normal
coordinates.

Locally, the connection $\nabla^{r}$ has the form

$\nabla^{r}_{\frac{\partial}{\partial_{\mathrm{x}_{k}}}}=\frac{\partial}{\partial_{\mathrm{x}_{k}}}+\Gamma_{k}^{S}+A^{0}_{k}+ra_{k},$

with $\Gamma^{S}_{k},A^{0}_{k}\in\mathcal{O}(\mathrm{x})$. Also,

$(\mathcal{D}^{r})^{2}=\nabla^{r*}\nabla^{r}+c(F_{0})+rc(da)+\frac{1}{4}\mathrm{Scal}$
(26)

Now we adapt the grading to count powers of $r$.

######  Definition 8.

The $r$-grading is defined as follows

$\displaystyle\mathrm{ord}_{rG}(\mathrm{x}_{j})=-1$
$\displaystyle\mathrm{ord}_{rG}(\frac{\partial}{\partial\mathrm{x}_{j}})=1$
$\displaystyle\mathrm{ord}_{rG}(r)=1$ (27)

and coefficients in $\mathrm{End}(S_{y}\otimes L_{y})$ have
$\mathrm{ord}_{rG}=0.$ The $r$-grading of a sum is the maximum of the
$r$-gradings of the summands.

Notice for example that $\mathrm{ord}_{rG}(ra_{k})\leq 1$ since the 1-form
$a=a_{j}d\mathrm{x}^{j}$ might not vanish at $\mathrm{x}=0.$ The proof of the
following result is almost identical to that of theorem 3.

######  Theorem 9.

Given two differential operators $A,B$ written formally as

$\displaystyle A$
$\displaystyle=\sum_{l,I,J}a_{l,I,J}r^{l}\mathrm{x}^{I}\partial_{\mathrm{x}}^{J}$
$\displaystyle B$
$\displaystyle=\sum_{k,S,T}b_{k,S,T}r^{k}\mathrm{x}^{S}\partial_{\mathrm{x}}^{T},$

with $a_{l,I,J}$ and $b_{k,S,T}$ in $\mathrm{End}(S_{y}\otimes L_{y})$
independent of $r.$ then

$\mathrm{ord}_{rG}(A\circ B)\leq\mathrm{ord}_{rG}(A)+\mathrm{ord}_{rG}(B).$

We have that
$\mathrm{ord}_{rG}(\nabla^{r}_{\frac{\partial}{\partial_{\mathrm{x}_{k}}}})\leq
1$, $\mathrm{ord}_{rG}(\mathcal{D}^{r})\leq 1$ and
$\mathrm{ord}_{rG}((\mathcal{D}^{r})^{2})\leq 2$. Similarly, it follows from
(9) that $\mathrm{ord}_{rG}(\Theta_{0})=0$ and
$\mathrm{ord}_{rG}(\Theta_{j})\leq 2j.$ The summands in $\Theta_{j}$ of
$\mathrm{ord}_{rG}=2j$ that do not vanish on the diagonal will have $j$ as the
highest possible exponent of $r.$

From the expansion (8) we get

$e^{-\frac{t}{r}(\mathcal{D}^{r})^{2}}(y,y)=k_{t/r}(y,y)\sim\frac{r^{n/2}}{(4\pi
t)^{n/2}}\sum_{j=0}^{\infty}\frac{t^{j}}{r^{j}}\Theta_{j}(y,y),$ (28)

which implies that the highest possible exponent of $r$ in (22) is $n/2.$ This
exponent can be reached only using the terms of highest $\mathrm{ord}_{rG}$ in
each $\Theta_{j}$. Therefore, we need to find the formula for the heat kernel
of

$H_{r}=-(\frac{\partial}{\partial\mathrm{x}_{k}}+ra_{k})^{2}+rc(da)$ (29)

Mehler’s formula gives

$\frac{1}{r^{n/2}}e^{-\frac{t}{r}H_{r}}=\frac{1}{(4\pi
t)^{n/2}}\sqrt{\det\dfrac{tA_{y}}{\sinh tA_{y}}}e^{-tc(da)}+o(r^{-1/2})$ (30)

Integrating (30) over $Y$, it follows (see [7, page 859]) that this expression
is equivalent to the one written in (25).

## References

  * [1] Berline, N. Getzler, E. Vergne, M. Heat Kernels and Dirac Operators. Grund. Math. Wiss. Band 298 (1992) Springer-Verlag, Berlin.
  * [2] Bismut, J. Leveau, G. Complex Immersions and Quillen Metrics. Inst. Hautes Études Sci. Publ. Math, 74 (1991).
  * [3] Getzler, E. Pseudodifferential Operators on Supermanifolds and the Index Theorem. Comm. Math. Phys, 92 (1983) pp. 163-178.
  * [4] Ma, X. Marinescu, G. Holomorphic Morse Inequalities and Bergman Kernels. Birkhäuser (2007).
  * [5] Roe, J. Elliptic Operators, Topology and Asymptotic Methods. Chapman and Hall/CRC, second edition (1998).
  * [6] Rosenberg, S. The Laplacian on a Riemannian Manifold. Cambridge Univ. press, London Math. Soc. Student Texts, 31 (1997).
  * [7] Savale, N. Asymptotics of the Eta Invariant. Comm. Math. Phys, 332 (2014) pp. 847-884.


# Delegation to autonomous agents promotes cooperation in collective-risk
dilemmas

Elias F. Domingos
AI Lab
Vrije Universiteit Brussel
1050 Brussels, Belgium
<EMAIL_ADDRESS>
&Inês Terrucha
IDLab
University of Ghent
B-9052 Ghent, Belgium
<EMAIL_ADDRESS>
&Rémi Suchon
ECARES
Université Libre de Bruxelles
1050 Brussels, Belgium
<EMAIL_ADDRESS>
Jelena Grujić
AI Lab
Vrije Universiteit Brussel
1050 Brussels, Belgium
<EMAIL_ADDRESS>
&Juan C. Burguillo
Department of Telematic Engineering
University of Vigo
36310 Vigo, Spain
<EMAIL_ADDRESS>
Francisco C. Santos
INESC-ID & Instituto Superior Técnico
Universidade de Lisboa
IST-Taguspark
2744-016 Porto Salvo, Portugal
<EMAIL_ADDRESS>
&Tom Lenaerts
MLG
Université Libre de Bruxelles
1050 Brussels, Belgium
<EMAIL_ADDRESS>

###### Abstract

Home assistant chat-bots, self-driving cars, drones or automated negotiations
are some of the several examples of autonomous (artificial) agents that have
pervaded our society. These agents enable the automation of multiple tasks,
saving time and (human) effort. However, their presence in social settings
raises the need for a better understanding of their effect on social
interactions and how they may be used to enhance cooperation towards the
public good, instead of hindering it. To this end, we present an experimental
study of human delegation to autonomous agents and hybrid human-agent
interactions centered on a public goods dilemma shaped by a collective risk.
Our aim to understand experimentally whether the presence of autonomous agents
has a positive or negative impact on social behaviour, fairness and
cooperation in such a dilemma. Our results show that cooperation increases
when participants delegate their actions to an artificial agent that plays on
their behalf. Yet, this positive effect is reduced when humans interact in
hybrid human-agent groups. Finally, we show that humans are biased towards
agent behaviour, assuming that they will contribute less to the collective
effort.

##  1 Introduction

Intelligent autonomous machines/agents are already present at many levels in
our daily lives. Examples can be found in a diverse range of domains:
autonomous vehicles [46, 37], fraud detection [36, 34, 10], personal home
assistants [24, 28], call centers [3], drones [4], recommendation systems [45]
or educational support systems [40, 23]. These intelligent agents facilitate
many of our individual needs while also influencing different aspects of our
social lives. Yet, their presence in social contexts raises questions about
the role they play in improving or diminishing our willingness to help others
or to collaborate in achieving some common future goals. How will human
behavior be changed once interactions are mediated by autonomous social
agents? Are they used to promote selfish or cooperative actions? Does
delegating decisions to agents reduce the sense of individual responsibility?
Can agents actually induce cooperation in humans systems, especially when
there is a conflict-of-interest between what’s good for oneself or good for
the group?

In this manuscript, we address some of these questions experimentally, using a
game-theoretical perspective. These behavioural economic experiments
investigate how human decision-making is affected when people i) delegate
their actions to one of a number of pre-defined autonomous agents, provided by
third parties and ii) interact in a hybrid human-agent group where
participants do not know who is an agent and who is a real person. In the
first context, our experiment also examines if having the capacity to "code"
or customise the agent with one’s own norms affects the observed behavior.
While this provides additional insight into the effect of delegation on
decision-making, it also provides insight into how people believe the agents
should behave when they are acting on their behalf.

In all experiments — delegation, programming of the agent, and hybridization —
we adopt the Collective-Risk Dilemma (CRD) [32, 38, 42, 44, 13, 25, 22], a
public goods game with uncertain and delayed returns. The CRD abstracts social
decision-making problems such as climate action, abuse of antibiotics or
collective hunting, where participants aim to avoid a future loss, rather than
to seek a benefit. Notwithstanding these interesting examples from human life,
collective risk situations are not exclusive to settings where only humans are
involved. There is an increasing number of examples of social cases that
involve autonomous agents and that involve different degrees of collective
risk. Examples range from self-driving cars to conversational (chat) bots in
call centers.

In the CRD, a group of human participants are confronted with a choice:
contribute sufficiently (cooperate) over several rounds to achieve a
collective target, ensuring that the group benefit is achieved, or contribute
insufficiently (defect) and assume that others will make the contributions to
reach the goal, and thus, aim to maximise one’s personal gains. Concretely, in
our setup of the CRD, participants, interact in groups of $6$, where each
member is required to contribute either $0$, $2$ or $4$ Experimental Monetary
Units (EMUs) in every round to a public account out of their private endowment
($40$ EMUs). The experiment consists of 10 rounds and, in the end, if the sum
of the total contributions of all players is higher or equal than a collective
target of $120$ EMUs, then all players will keep the remainder of their
endowment. Otherwise, all players lose their remaining endowment with a
probability of $90\%$, which represents the collective risk of the dilemma and
thus gain nothing.

In earlier experiments, Milinski et al. [32] have evaluated how different risk
probabilities affect human decision-making in the CRD, finding that
participants only start to contribute when the risk is high ($p=0.9$).
Additionally, Fernández Domingos et al. [22] repeated the treatment with
$90\%$ of risk of Milinski et al. [32], adding two additional treatments that
investigate how uncertainty about the number of rounds available to reach the
collective target affected participants’ behaviour. They found that this
timing uncertainty increases the polarisation in the population while
prompting earlier contributions and reciprocal behaviors in successful groups,
which could indicate that under such uncertainty, signaling the willingness to
contribute earlier could be key for success. We use the results of their first
treatment as control for the experiment presented in this manuscript.

Now, if no humans take part in a CRD, but instead, they select artificial
agents to play the game for them, what would be the outcome? To test this, we
performed a first treatment in which participants have to delegate their
actions during the experiment to an autonomous agent that they select from a
small set of possibilities. This experiment examined, on the one hand, which
agents are selected and, on the other hand, whether the group of selected
agents is more or less successful than humans. In a second treatment, all
participants get the same template-agent and are asked to _customise_
(rudimentary programming) the agent so it follows an algorithm that represents
their beliefs/preferences on how the game should be played. At the same time,
we verified what preferences people have concerning these agents, i.e.,
whether they prefer to play themselves (or not).

In a third treatment, we examine what happens when the group consists of a
combination of humans and agents: is there any difference in the success rate
compared to a CRD with only humans or only agents? Again the question depends
on the effect of different agent combinations in the group, as different
choices are possible. Here we asked what would happen in the “best” case,
thus, selecting the agents programmed by participants from successful groups
from the previously mentioned customisation experiment (see also details
below). Moreover, we evaluate whether participants think agents made more or
less contributions than human participants.

The remainder of the paper is organised as follows. In Section 2 gives an
overview of the experimental and theoretical literature about delegation and
hybrid interactions with autonomous agents. Section 3 introduces our
experimental design and in Section 4 we present and discuss our results.
Finally, we summarise our final conclusions in Section 5.

##  2 Background

The question of how the presence of autonomous agents may affect human social
interactions has been explored from sociological, strategical and
psychological perspectives [1, 31, 7, 16, 17, 43, 33, 39]. In the context of
behavioural economics, where strategic interactions between individuals is
analysed, March [31] finds, after comparing more than 90 experimental studies,
that humans tend to act in a more self-interested or selfish manner when
computer players are present, and, often, they are able to exploit such
players. Also, Cohn et al. [11] show that individuals tend to cheat about
three times more when interacting with agents instead of another human, and
that dishonest individuals prefer to interact with agents. Although, as
artificial intelligence (AI) evolves, the opposite situation may also occur,
i.e., intelligent agents may exploit humans. Camerer [7] argues that
behavioural experiments are necessary to understand and predict how this may
happen.

Initial results by de Melo et al. [17] have shown experimentally that
delegating to programmable autonomous agents changes the way in which
participants approach a social dilemma and is able to promote the evaluation
of long-term consequences of choices, which in their case increases
cooperation. Also, in another experimental study, de Melo et al. [16] find
that participants behave more fairly when acting through autonomous agents in
comparison to when human participants directly interact among themselves. They
also indicate that when participants program the agents they tend to rely on
social norms, which could explain the increase in fairness.

Introducing agents, programmable/customisable by users to represent their
interests in social settings has a number of benefits. The literature commonly
refers to these agents as _peer-designed agents_ [9, 8, 19, 30], and they can
be used to reduce costs in the design and testing of mechanisms that involve
human decision-makers, such as automated negotiations and mechanism evaluation
[29, 19], agent-based simulations [8] or testing experimental hypothesis in
game-theoretic scenarios. At the same time, those applications assume that the
peer-designed agents truly represent their designer strategy/behaviour.
However, this may not always be true and may depend on the context of the
decision-making scenario.

For instance, Grosz et al. [26], Elmalech et al. [20] and Manistersky et al.
[30] find that, in a negotiation (or trading) setting, participants design
agents closer to rational game-theoretic models, instead of more fair or
cooperative agents, which contrast with human behaviour observed in direct
interactions [41, 2] and the increased fairness observed by de Melo et al.
[17]. Also, other authors find that programmed agents can increase the ability
of participants to coordinate when negotiating [35], which can be an useful
trait for our case study in the CRD.

When it comes to hybrid interactions between humans and autonomous agents, as
previously mentioned, the game-theoretic literature indicates that humans tend
to act more selfishly in those scenarios [31]. However, de Melo and Terada
[14] show experimentally that, by adding visual emotional and cultural cues to
the agents, this cooperation barrier against the agents can be surpassed.
Concretely, the authors find that while cultural cues cause similar bias as
those observed against humans, when agents express emotional cues,
participants cooperated with them just as much as against other humans. This
research also hints that trust is a very important factor in human-agent
interactions. In this line, Andras et al. [1] suggests that a framework of
trust between humans and autonomous agents must be developed. Moreover, it
also prompts further research on how such trust currently affects hybrid
strategic interactions. Han et al. [27] argues that trust-based strategies can
be very effective in human-agent interactions, particularly when there is a
lack of transparency about the agents’ strategies.

In general, experimental results appear to be dependent on the context of the
strategic interaction, generating two opposing hypotheses: i) interaction with
or through autonomous agents affects social preferences in such a way that it
promotes more selfish behaviours and thus decreases cooperation; ii)
interacting through and with autonomous agents may eliminate or reduce
negative effects of emotions such as revenge or _fear of betrayal_ [12, 18, 6,
21, 5], favoring pro-social behaviours and thus the increase cooperation.

##  3 Experimental design

To explore experimentally the questions exposed in the introduction, we
designed three experimental treatments that test how the presence of
autonomous agents influence participant behaviour. These treatments are
compared to a control experiment in which only humans participate.

In all treatments, participants play a CRD in a group of $6$ members for 10
rounds with a collective target of $T=120$ EMUs. Each participant starts with
a private endowment of $E=40$ EMUs and will lose their remaining endowment at
the end of the game with probability $p=0.9$ if the joint contributions of the
group do not reach the collective target.

###  3.1 Control treatment with only humans

In the control experiment, human participants from each session are randomly
sampled in groups of $6$. All their decisions are anonymous, i.e., neither the
other participants nor the experimenters know the identity of the participant
making the decisions. The experiment is held in a laboratory and participants
interact with each other through a computer interface. In each round, each
participant is able to observe their own contribution in the previous round
and the contributions of the other participants, thus, allowing them to adopt
conditional decisions. At the end of the experiment, participants are shown
the result of the experiment. In case their group did not reach the target,
they are shown the exact result as a function of the $p$, i.e. a randomly
generated value determines whether they lose or win their remaining endowment.
Finally, they have to complete a short survey that elicits their motives
during the experiment. The full description of the results of this treatment
are presented in [22].

###  3.2 _Delegate_ treatment (T1)

Before the experiment starts, participants are requested to choose one of 5
types of artificial agents to play the game in their place (see Figure 1c).
Thus, in this treatment, participant make a single decision in the experiment.
For convenience, we attribute names to the available agent behaviours, i.e.,
always-0, always-2, always-4, compensator and reciprocal. However, during the
experiment, agents were only presented by an alphabet letter (e.g., A, B,
etc.) and there was never any framing related to the fairness of the agent’s
behaviours. Afterwards, they are able to observe the decisions of their agent
and the other agents in the group throughout the game. They can also observe
the content of the public and private accounts. At each round, participants
have a maximum of 10 seconds to observe the information in the screen. Once
the game ends, the result is shown on the screen, including information about
the participant’s final payoff. Finally, participants are requested to
complete a short questionnaire that helps elucidate their decision to select a
particular agent and how they experienced the performance of their agent.

Figure 1: Representation of agent behaviour. Panel a) classifies the behaviour
of agents in relation to how much they contribute in response to a stimuli
that is above or below a certain personal threshold $\tau_{p}$. When an agent
contributes the same positive amount for stimuli below or above the threshold,
we say that it cooperates. When it contributes more for values below (above)
the threshold, we say that the agent compensates (reciprocates). Finally, when
the agent does not contribute, we say that it defects. Panel b) shows a
general structure for the description of the agent’s behaviour. An agent has 5
parameters that consist on an initial contribution ($a_{0}$), a personal
threshold for the external stimuli ($\tau_{p}$), and the contributions to be
made when the external stimuli is above ($a_{a}$), equal ($a_{m}$), or below
($a_{b}$) $\tau_{p}$. Panel c) represents the pool of agents that can be
selected in the _delegate_ treatment. Participants may select between
cooperative agents (e.g., always-2, always-4), conditional agents (e.g.,
reciprocal, compensator) and defecting agents (e.g., always-0), however, these
names are not shared with them, and agents are named with letters from the
alphabet (e.g., A, B, etc.).

####  3.2.1 Description of the agents

The first three agents (_always-0_ , _always-2_ , _always-4_) display
unconditional behaviours, i.e., they contribute in every round 0, 2, or 4 EMUs
respectively. The last two exhibit conditional behaviours, which respond to
external stimuli, in this case, the total contributions of the group members
in the previous round ($a_{-i}(t-1)$), in function of a personal threshold
$\tau_{p}$, which defines a trigger for behavioural change in the autonomous
agents. Agents compare external stimuli to this threshold and adapt their
contribution levels depending on the comparison result.

In this treatment, we assume $\tau_{p}=10$ EMUs, representing the total amount
of contributions in one round made by the other participants. We call this a
locally fair amount since if participants contribute this amount in every
round and you also contribute $2$ in every round then the threshold of $120$
EMUs would be reached by the end of the game and everyone will have the same
gains to take home. Given this threshold $\tau_{p}$, _Compensators_ only
contribute if $a_{-i}(t-1)\leq\tau_{p}$. In contrast, _reciprocal_ agents only
contribute if $a_{-i}(t-1)\geq\tau_{p}$. All autonomous agents stop donating
once the collected contributions reach (or surpass) the collective target.
These agent types emulate the behaviors identified in the _human_ control
treatment [22].

###  3.3 Customise treatment (T2)

In the customization treatment, participants are requested to configure an
artificial agent that will act in their place during the experiment. Again, in
this treatment, participants make a single decision in the experiment. Each
artificial agent can be customised in such a way that it can adapt its action
to the actions of the other agents within the group, i.e., participants make a
conditional agent. To do this, participants are requested to configure the
values of 5 parameters (see Figure 1b) that define an agent’s behaviour:

  * •

Threshold ($\tau_{p}\in[0..20]\text{ in steps of 2}$): is an integer that will
be compared to the total contribution of the other group members in the
previous round, i.e., the agent will select one of the three following actions
based on whether the contributions of the other members of the group in the
previous round ($a_{-i}(t-1)$) are greater, equal or lesser than.

  * •

Initial action ($a_{0}\in{0,2,4}$): the action the agent will take in the
first round of the game.

  * •

Action above ($a_{a}\in{0,2,4}$): the action the agent will take if
$a_{-i}(t-1)>\tau_{p}$.

  * •

Action equal ($a_{m}\in{0,2,4}$): the action the agent will take if
$a_{-i}(t-1)=\tau_{p}$.

  * •

Action below ($a_{b}\in{0,2,4}$)): the action the agent will take if
$a_{-i}(t-1)<\tau_{p}$.

Allowing participants to configure their agent, effectively increases the
behavioural space that can be displayed in the CRD in comparison to the
delegate treatment. Concretely, there is a maximum of $10*3^{4}=810$ possible
behavioural combinations in which an agent can be configured. Thus, if we
observe an increase in collective success in both treatments, we can rule out
that this result is caused by a limited number of choices, and instead we may
conclude that it is an effect of delegating to autonomous agents. Finally, in
order to assess the level of trust in the delegation system, we ask
participants once the experiment has finished, whether, if given the option in
another experiment, they would decide to delegate/customise again, or play the
game themselves.

###  3.4 Nudge treatment (T3)

In the hybrid experiment, we performed the same setup as described in the
human control treatment (see Section 3.1) . Except that in this case, each
group is formed by 3 human participants and 3 artificial agents. Participants
are told that the agents have been designed to adopt a human-like behaviour in
the CRD experiment: we selected the agents’ behaviours from the pool of agents
that were part of successful groups in the customise treatment (see Section
3.3)). Since we select only agents programmed by participants from successful
groups (of the customise treatment), our hypothesis is that those agents may
be able to _nudge_ human participants into cooperative behaviors.

At the end of the experiment, we test whether participants were able to
discern which members of the group were human and which were not. The
participants are not informed who the agents are in the session in order to
ensure that they are just responding to the actions and not the fact that the
action is performed by a human or an agent.

###  3.5 Sample and collected data

A total of $246$ participants were part of this experiment, of which $174$
participated in the 3 treatments that involve autonomous agents. The
experiment was carried out at the _< removed for reasons of anonymity>_. The
population was mainly composed of university students between the ages of $19$
and $33$, of which $49.4\%$ were female. The _delegate_ treatment had $n=90$
participants divided into $15$ groups of $6$ participants each. The
_customise_ treatment had $n=48$ participants and $8$ groups. Finally, the
nudge treatment had a total of $n=48$ participants and 16 hybrid groups.
Moreover, the _human_ control treatment was part of another experiment
presented in [22] and consisted of $n=72$ participants divided into $12$
groups of $6$ participants. Due to the COVID-19 situation, we were unable to
perform more sessions of the customise and nudge. Nevertheless, our current
data already hints towards interesting phenomena.

For all treatments, we stored the result of the experiment, the configuration
over all rounds and the response to a final questionnaire. In delegate (and
customise) treatment we also stored, for each participant, the selected
(customised) agent and the time taken to select it. For the nudge treatment,
we collected the contribution at each round, the actions of the other members
of the group in the previous round, the content of the private and public
accounts at each round, the predictions of the total content of the public
account after each round, and the time taken to make each decision. We also
stored the responses to a follow-up survey that participants had to complete
once the experiment was finished. In this regard it is important to mention
that $1$ participant has been excluded from the survey results in the delegate
treatment, since the participant did not complete the survey.

Figure 2: Group achievement and average payoff. The group achievement obtained
in the _humans_ control treatment (black: $n=12$) is compared to the results
of the _delegate_ (red: $n=15$), _customise_ (orange: $n=8$) and _nudge_
(yellow: $n=16$) treatments. There is an increase in groups that achieve the
target on the _delegate_ and _customise_ treatments, however this trend is not
maintained in the _nudge_ treatment.

##  4 Results and Discussion

The fraction of successful groups (group achievement, $\eta$) is an important
indicator of the cooperative response in the CRD. In the treatment with only
_humans_ , $66.7\%$ of the groups successfully achieved the collective target,
despite the high level of risk ($p=0.9$). In a previous CRD experiment,
Milinski et al. indicate an even lower result ($\eta=0.5$). In contrast,
Figure 2 shows that the fraction of successful groups increases in the
_delegate_ and _customise_ treatments ($87\%$ and $87.5\%$ respectively),
which indicates that delegation has a positive effect on group success.
However, this trend does not persist in the _nudge_ treatment, where
participants interact with agents and not through them.

Figure 3: Agent choice bias and participant response. Panel (a) shows the
distribution of agent behaviors chosen by the participants (right of the black
line) in the _delegate_ treatment ($n=89$) in comparison to the expected
values predicted by the model described in [22] (left of the black line). The
observed values indicate that there is a bias toward behaviors that are
beneficial for the success of the group. Panel (b) shows the percentage of
participants that would choose to delegate/customize an agent to play in their
place if they were given the option to participate again in the experiment by
themselves or to choose an agent. These results show that most participants
would rather participate in the experiment by themselves, despite the higher
rates of group success in comparison to the experiments with only humans.
Nevertheless, there is an increase in participants that would delegate again
in the _customise_ experiment ($n=48$). This could indicate that allowing the
customisation of artificial agents in CRD systems could be a potential
relevant design feature to increase their penetration across users. Panel (c)
shows the percentage of participants who consider that humans had made the
greatest contribution effort during the _nudge_ treatment, compared to the
average contribution difference of agents with respect to humans in a group.
These results show that while $80\%$ of the human players from failed groups
($n=15$) believed humans contributed more during the experiment, artificial
agents had marginally contributed more ($+0.33\%$). Such a result indicates a
strong perception bias that attaches guilt of failure to the artificial
agents. This perception decreases in participants from successful groups,
where only $48.48\%$ believe humans had put most of the effort ($n=33$), yet
the extra effort made by the agents is only and $8.48\%$ higher than that of
the human players.

Also, if we look at the distribution of selected agent behaviours (see Figure
3a, right of the black line) in the delegate treatment, we can observe that
always-2 and compensator were the behaviours selected more frequently. This is
in agreement to the predictions of the model of the experimental results of
the _humans_ control treatment (left of the black line) described in [22].
Yet, the predicted and observed results for the _always-0_ , _always-4_ and
_reciprocal_ agents differ, with an increased preference for cooperative
strategies (_always-4_ and _reciprocal_) and a decrease in the fraction of
_always-0_. We attribute this difference to a delegation bias towards
behaviours that are beneficial to the collective, which is corroborated by the
answers on the surveys (most participants explain that their motivation was to
maximise the probability of group success as opposed to their personal
benefit).

The literature on behavioural economics [12, 18, 6, 21, 5] theorises that when
interacting in a group of humans, participants tend to be afraid of being
betrayed or cheated by the other participants. This fear of betrayal, or
_betrayal aversion_ , is reduced when humans act through autonomous agents
[15, 16, 17]. Additionally, the chosen/customised agent cannot be changed
during the experiment. Therefore, participants are forced to think about the
future consequences of their choice, transforming delayed rewards into
immediate ones. These several factors play an important role in the observed
increase of cooperation.

Moreover, at the end of the delegation experiments, participants were asked
whether, if given the option to play again, they would choose to delegate once
more to an artificial agent, or to play the game themselves. Figure 3b shows
that the majority of participants in the _delegate_ and _customise_ treatments
would prefer to play the game themselves, even though the results from the
_humans_ treatment indicate that this would lead to lower levels of collective
success. This choice suggest a reticence to give away agency, and a preference
for retaining control over the actions at each round. Yet, in the _customise_
treatment, there was an increment in the fraction of participants that would
delegate, suggesting that the ability to customise/configure the agent,
increases participants’ trust in it, which is a relevant feature when
designing delegation systems. Therefore, future research should explore how
delegation and customization should be combined to achieve both cooperation
and user satisfaction.

Finally, regarding the _nudge_ treatment in which humans interact in hybrid
groups with autonomous agents, our current data shows that there is a
significant proportion of participants that believe agents make less effort
than human participants, even though this is not true. Concretely, in Figure
3c, we show that the majority of participants from failed groups ($80\%$,
$n=15$) consider that humans have made more contributions than agents, when in
reality artificial agents have marginally contributed more ($+0.33\%$). At the
same time, in successful groups, $48.48\%$ of participants ($n=33$) still
believe that humans are making more effort than agents, while in this case
agents have made $8.48\%$ more contributions than humans. This implies that
there exists a negative perception towards artificial agents, i.e.
participants associate a higher degree of selfishness to them than to humans.
On the other hand, and however we would require another experiment to prove
this hypothesis, a hybrid group seems to focus human attention on the agents,
indirectly producing a similar reduction in betrayal aversion to what has been
observed in the delegation treatments.

##  5 Conclusions

Previous research has shown that people, when delegating to other humans, tend
to choose those who act more selfishly [21, 5]. In contrast, observations
throughout this experiment suggested that people prefer to delegate their
decisions to agents that benefit the collective, thus increasing cooperation.
This has also been observed in previous experiments with autonomous agents
[17]. Yet, subjects still prefer to retain control of their actions. However,
being able to customise their agent, increased the fraction of participants
that were willing to delegate. But, when interacting with these customised
agents, subjects in hybrid groups tended to assign the responsibility of
failure to the autonomous agents. Therefore, further investigation should
address the interplay of social expectations towards autonomous agents, when
delegating and in hybrid interactions, with the ability to embed preferences
into the agents through customisation.

## Broader Impact

The experimental study presented in this manuscript offers insights into the
impact of delegation towards autonomous agents and hybrid collective systems.
Our work shows that the use of artificial delegates, in cases where human
fears like betrayal aversion may negatively influence the outcome, can improve
the chances of a positive outcome. Defining the decision-making process for
future gains upfront and explicitly makes participants think about what they
want, as opposed to emotionally responding to cues coming from other
participants in the games. Yet, still, it is not enough to reach a $100\%$
success. Communicating what agent a participant selects or informing others
about the customisation of a participant’s agent, may ensure that this last
hurdle is overcome, just as in the use of pledges [42, 13]. The similarity of
the CRD with many real-world situations also ensures that the conclusions we
draw from these experiments have a wide impact, hopefully helping to achieve
fundamental understandings.

The current results have natural limitations associated with the the
demographics of the participants pool as well as its size. As it is often the
case in this type of studies, it would be meaningful to explore how other age-
groups would deal with delegation to autonomous systems and to expand further
the number of sessions for the the nudge and customisation treatments. We have
indications that children respond differently. Such expansions will allow us
to remove biases in the experimental results, helping us to find further
support for the observations we have made. Moreover, although our experiments
hint towards a relationship between trust and customisation, a different
experiment is required to provide a deeper understanding of the mechanisms
behind this result. Finally, our study has focused on a scenario of high
collective risk. We believe it would be important to observe whether the
levels of cooperation and human biases remain unchanged in scenarios of
intermediary or lower level of risk, or whether participants’ preferences
towards intelligent agents are risk dependent.

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# Spectral Theorem for a bounded self adjoint operator on a Bicomplex Hilbert
space

Akshay S. RANE111Department of Mathematics, Institute of Chemical Technology,
Nathalal Parekh Marg, Matunga, Mumbai 400 019, India, email :
<EMAIL_ADDRESS>and Mandar THATTE222Department of Mathematics,
Institute of Chemical Technology, Nathalal Parekh Marg, Matunga, Mumbai 400
019, India, email<EMAIL_ADDRESS>
###### Abstract

In this paper, we shall consider the notion of bicomplex inner product and
define bicomplex Hilbert space. We shall define $L^{2}[a,b]$ where the
functions take bicomplex values. We shall prove the Theorem for a bounded self
adjoint operator on a bicomplex Hilbert space which is not compact.

Key Words : Bi complex inner product, Bicomplex Hilbert module, Self Adjoint
operator, $L^{2}[a,b]$, Spectral theorem.

AMS subject classification : 16D10, 30G35, 46C05, 46C50.

##  1 Introduction

The notion of Bicomplex numbers and hyperbolic has been introduced in [7].
They have also discussed the idempotent representation of bicomplex numbers.
Fundamental theorems in functional analysis like the Baire Category theorem,
open mapping theorem, closed graph theorem, uniform boundedness principle for
modules over bicomplex/hyperbolic numbers have been proved in [8]. Zabreiko’s
lemma and as a consequence the fundamental theorems have been proved in
[9].The notion of bicomplex inner product and definition of adjoint operator
has been dicussed in [3] and [2]. The notions of spectrum and resolvent
operator in case of a bicomplex Banach module is introduced in [6]. The
spectral decomposition theorem for a compact self adjoint operator on a
bicomplex Hilbert space has been proved in [4]. Spectral theorem for a self
adjoint on the usual Hilbert space is well known. One such reference is[1]. In
this paper, we shall first define the space $L^{2}[a,b]$, where the functions
defined on $[a,b]$ takes bicomplex values. We define the inner product on this
space consistent as what is done for $l_{2}$ in [5]. Later we shall prove the
spectral theorem for a bounded self adjoint operator on a Bicomplex Hilbert
space which is not compact. The set of bi complex numbers is defined as

$\mathbb{B}\mathbb{C}=\\{Z=z_{1}+z_{2}j|z_{1},z_{2}\in C(i)\\}$

where $i$, $j$ are such that $ij=ji$ and $i^{2}=j^{2}=-1.$ The set
$\mathbb{B}\mathbb{C}$ forms a Ring under the usual addition and
multiplication. The product of imaginary units $i$ and $j$ defines a
hyperbolic unit $k$ such that $k^{2}=1.$ Since $\mathbb{C}$ contains two
imaginary units whose square is -1 and one hyperbolic unit whose square is 1,
we consider three conjugations for bi complex numbers.

  1. 1.

$\overline{Z}=\overline{z_{1}}+\overline{z_{2}}j$ (bar-conjugation).

  2. 2.

$Z^{+}=z_{1}-z_{2}j$ ($+$ \- conjugation).

  3. 3.

$Z^{*}=\overline{z_{1}}-\overline{z_{2}}j$ (*- conjugation).

The set $\mathbb{D}$ of hyperbolic numbers is defined as

$\mathbb{D}=\\{h=h_{1}+kh_{2}|h_{1},h_{2}\in\mathbb{R}\\}.$

The set $\mathbb{D}$ is a ring and module over itself. Also,

$\mathbb{D^{+}}=\\{h=e_{1}a_{1}+e_{2}a_{2}|a_{1},a_{2}\geq 0\\}$

is the set of all positive hyperbolic numbers. We define $e_{1}$ and $e_{2}$
as

$e_{1}=\frac{1+k}{2},e_{2}=\frac{1-k}{2}$

and satisfy following properties

$e_{1}^{2}=e_{1},e_{2}^{2}=e_{2},e_{1}+e_{2}=1,e_{1}e_{2}=0,e_{1}^{*}=e_{1},e_{2}^{*}=e_{2}$

In fact $e_{1}$ and $e_{2}$ are the idempotent basis of bi complex numbers and
any bi complex number can be uniquely written as

$w=z_{1}+z_{2}j=w_{1}e_{1}+w_{2}e_{2}$

where,

$w_{1}=z_{1}-z_{2}i,w_{2}=z_{1}+z_{2}i.$

Also,

$|w|=\frac{\sqrt{|w_{1}|^{2}+|w_{2}|^{2}}}{\sqrt{2}}.$

Let X be a $\mathbb{B}\mathbb{C}$ module then its idempotent representation is

$X=e_{1}X_{1}\oplus e_{2}X_{2}=e_{1}{(e_{1}X)}\oplus e_{2}{(e_{2}X)}.$

##  2 Bicomplex Inner Product

We now consider the bicomplex inner product.Let $X$ be $\mathbb{B}\mathbb{C}$
module. A map

$\langle.,.\rangle:X\times X\rightarrow\mathbb{B}\mathbb{C}$

is said to be a $\mathbb{B}\mathbb{C}$ inner product if it satisfies following
properties for all $x,y,z\in X$ and $\alpha\in\mathbb{B}\mathbb{C}$:

  1. 1.

$\langle x,y+z\rangle$ = $\langle x,y\rangle+\langle x+z\rangle$

  2. 2.

$\langle\alpha x,y\rangle$ = $\alpha\langle x,y\rangle$

  3. 3.

$\langle x,y\rangle$= $\langle y,x\rangle^{*}$

  4. 4.

$\langle x,x\rangle\in\mathbb{D^{+}}$ and $\langle x,x\rangle=0$ if and only
if $x=0.$

######  Definition 2.1.

A $\mathbb{B}\mathbb{C}$ module $X$ endowed with a bi complex inner product
$\langle.,.\rangle$ is said to be a $\mathbb{B}\mathbb{C}$ inner product
module.

Suppose $X_{1}$ and $X_{2}$ over $C[i]$ are equipped with the inner product
$\langle.,.\rangle_{1}$ and $\langle.,.\rangle_{2}$ and corresponding norms
$\|.\|_{1}$ and $\|.\|_{2}.$ The formula $\langle.,.\rangle_{X}=e_{1}\langle
x_{1},w_{1}\rangle_{1}+e_{2}\langle x_{2},w_{2}\rangle_{2}$ defines bi complex
inner product on the bi complex module $X.$ The hyperbolic norm can be
introduced on the inner product $\mathbb{B}\mathbb{C}$ module given by

$\|x\|_{\mathbb{D}}=e_{1}\|x_{1}\|_{1}+e_{2}\|x_{2}\|_{2}.$

It is also possible to introduce a real valued norm on Bicomplex module by
setting

$\|x\|_{X}^{2}=\frac{1}{2}(\|x_{1}\|_{1}^{2}+\|x_{2}\|_{2}^{2}).$

The relation between the two norms is

$|\|x\|_{\mathbb{X}}|=\|x\|_{\mathbb{D}}$

######  Definition 2.2.

A $\mathbb{B}\mathbb{C}$ inner product module $X$ is said to be a bi complex
Hilbert module if $X$ is complete with respect to the hyperbolic norm
generated by the inner product square.

Any function $f:[a,b]\rightarrow\mathbb{B}\mathbb{C}$ can be written as

$f(t)=x_{0}(t)+ix_{1}(t)+jx_{2}(t)+ijx_{3}(t)=$
$[(x_{0}(t)+x_{3}(t))+i(x_{1}(t)-x_{2}(t))]e_{1}+[(x_{0}(t)-x_{3}(t))+i(x_{1}(t)+x_{2}(t))]e_{2}=u(t)e_{1}+v(t)e_{2}$

where the functions $u,v:[a,b]\rightarrow\mathbb{C}=C[i]$ .

$|f(t)|^{2}=|x_{0}(t)|^{2}+|x_{1}(t)|^{2}+|x_{2}(t)|^{2}+|x_{3}(t)|^{2}.$

Let $L^{2}[a,b]$ denote the space of bi complex valued functions that are
square integrable Let $\mu$ denote the lebesgue measure on $[a,b].$

$\int_{a}^{b}|f(t)|^{2}d\mu(t)<\infty.$

The inner product

$\langle f,g\rangle=\int_{a}^{b}f(t)\overline{g(t)}d\mu(t)$

gives

$\langle
u,u\rangle=\int_{a}^{b}|x_{0}(t)+x_{3}(t)|^{2}+|x_{1}(t)-x_{2}(t)|^{2}d\mu(t)$

and

$\langle
v,v\rangle=\int_{a}^{b}|x_{0}(t)-x_{3}(t)|^{2}+|x_{1}(t)+x_{2}(t)|^{2}d\mu(t).$
$\frac{\|u\|_{2}^{2}+\|v\|_{2}^{2}}{2}=\int_{a}^{b}(|x_{0}(t)|^{2}+|x_{1}(t)|^{2}+|x_{2}(t)|^{2}+|x_{3}(t)|^{2})d\mu(t)=\|f\|_{X}^{2}$
$\|f\|_{X}=\sqrt{\frac{\|u\|_{2}^{2}+\|v\|_{2}^{2}}{2}}$

$\mathbb{B}\mathbb{C}$ module $X$ can also be viewed as a normed vector space
$X^{\prime}$ over $\mathbb{C}=C[i].$ (See [4].) $L^{2}[a,b]$ becomes a
bicomplex Hilbert space with respect to the norm of the associated vector
space $L^{2}[a,b]^{\prime}$ over $\mathbb{C}=C[i]$ is defined by

$\int_{a}^{b}|f(t)|^{2}d\mu(t).$

Note that

$L^{2}[a,b]^{\prime}=(e_{1}L^{2}[a,b])+(e_{2}L^{2}[a,b])$

$L^{2}[a,b]$. Let $H$ be a bicomplex Hilbert space. Let $T:H\rightarrow$ be a
bi complex linear operator over $H$. The adjoint of the operator is defined as

$\langle x,Ty\rangle=\langle T^{*}x,y\rangle$

for all $x.y\in H$ Then the adjoint operator $T^{*}$ for $H$ considered as a
bicomplex module is same as the adjoint of operator on $H^{\prime}$ that is
$H$ considered as $\mathbb{C}=C[i]$ vector space. $T$ is self adjoint if
$T^{*}=T.$ The bi complex space $L^{2}[a,b]$ is a $\mathbb{B}\mathbb{C}$
module.

##  3 Main Results

######  Definition 3.1.

Let X Bi complex Module. A map $T:X\rightarrow X$ is said to be linear if for
all $a,b\in X$, we have $T(a+b)=T(a)+T(b)$ and $T(\mu a)=\mu T(a)$ for
$\mu\in\mathbb{B}\mathbb{C}.$

We now prove the special case of Spectral Theorem for the case when the
bicomplex Hilbert space has a cyclic vector. A vector is said to be cyclic if
the closure of the span of the elements $\\{x,Tx,T^{2}x,\ldots,\\}$ is the
whole of $X.$

######  Theorem 3.2.

Let $H$ be a bi complex Hilbert module. Let $T$ be a linear operator on $H.$
Let $H=e_{1}H_{1}+e_{2}H_{2}$ be its idempotent decomposition. Suppose $H$ has
a cyclic vector then $H_{1}=e_{1}H$ and $H_{2}=eH$ also have a cyclic vector.

###### Proof.

Let $w$ be a cyclic vector in $H.$ Since $H=e_{1}H_{1}+e_{2}H_{2}$,
$w=e_{1}w_{1}+e_{2}w_{2}=e_{1}(e_{1}w)+e_{2}(e_{2}w).$
$\overline{Span\\{w,Tw,T^{2}w,\ldots\\}}=H.$
Claim: $H_{1}$ has a cyclic vector $e_{1}w.$ Let $z\in H_{1}$, so $z=e_{1}u$ ,
$u\in H.$ Since, $u\in\overline{Span\\{w,Tw.T^{2}w,...\\}}.$ So there exists a
sequence $S_{n}\in span\\{w,Tw,T^{2}w,...\\}$ such that $S_{n}\rightarrow
u=e_{1}(e_{1}u)+e_{2}(e_{2}u)=e_{1}u_{1}+e_{2}u_{2}.$ So,

$S_{n}=\sum_{k=0}^{m(n)}\alpha_{k}T^{k}(w)$

where
$\alpha_{k}=e_{1}\alpha_{k}^{{}^{\prime}}+e_{2}\alpha_{k}^{{}^{\prime\prime}}.$
Since

$T(w)=e_{1}T_{1}(e_{1}w)+e_{2}T_{2}(e_{2}w)$

so,

$T^{k}(w)=e_{1}T_{1}^{k}(e_{1}w)+e_{2}T_{2}^{k}(e_{2}w)$

for each $k\in N.$ Hence,

$S_{n}=\sum_{k=0}^{m(n)}\alpha_{k}^{{}^{\prime}}e_{1}T_{1}^{k}(e_{1}w)+\alpha_{k}^{{}^{\prime\prime}}e_{2}T_{2}^{k}(e_{2}w)=e_{1}\sum_{k=0}^{m(n)}\alpha_{k}^{{}^{\prime}}T_{1}^{k}(e_{1}w)+e_{2}\sum_{k=0}^{m(n)}\alpha_{k}^{{}^{\prime\prime}}T_{2}^{k}(e_{2}w).$
$=e_{1}S_{n}^{{}^{\prime}}+e_{2}S_{n}^{{}^{\prime\prime}}.$

Hence,

$|\|S_{n}-u\|_{\mathbb{D}}|^{2}=\|S_{n}-u\|^{2}=\frac{\sqrt{\|S_{n}^{{}^{\prime}}-u_{1}\|_{1}^{2}+\|S_{n}^{{}^{\prime\prime}}-u_{2}\|_{2}^{2}}}{\sqrt{2}}\geq
0.$

Since $\|S_{n}-u\|_{\mathbb{D}}^{2}\rightarrow 0$ we get
$S_{n}^{{}^{\prime}}\rightarrow u_{1}$ and
$S_{n}^{{}^{\prime\prime}}\rightarrow u_{2}.$ Thus,

$H_{1}=\overline{span\\{w_{1},Tw_{1},T^{2}w_{1},....\\}}$

and

$H_{2}=\overline{span\\{w_{2},Tw_{2},T^{2}w_{2},...\\}}.$

Therefore, $H_{1}$ has a cyclic vector $e_{1}w$ and $H_{2}$ has a cyclic
vector $e_{2}w.$ ∎

This following proposition is proved for the usual Hilbert space in Bhatia
[1].

######  Proposition 3.3.

Suppose $T$ is a self adjoint operator on the usual Hilbert space $H$ with a
cyclic vector $H.$ Then there exists a probability measure $\mu$ on the
interval $[-\|T\|,\|T\|]$ and a unitary operator $U$ from $H$ onto
$L^{2}[-\|T\|,\|T\|]$ such that $UTU^{*}=M$ the canonical multiplication
operator on $L^{2}[-\|T\|,\|T\|].$

We now have the following proposition.

######  Proposition 3.4.

Suppose $T$ is a self adjoint operator on the bicomplex Hilbert module $H$
with a cyclic vector $H$. Then there exists a probability measures $\mu_{1}$
and $\mu_{2}$ on the interval $[-\|T_{1}\|,\|T_{1}\|]$ and
$[-\|T_{2}\|,\|T_{2}\|]$ unitary operators $U_{1}$ from $H_{1}$ onto
$L^{2}[-\|T_{1}\|,\|T_{1}\|]$ and $U_{2}$ from $H_{2}$ onto
$L^{2}[-\|T_{2}\|,\|T_{2}\|]$ such that $UTU^{*}=M$, where
$U=e_{1}U_{1}+e_{2}U_{2}$, $T=e_{1}T_{1}+e_{2}T_{2}$ and
$M=e_{1}M_{1}+e_{2}M_{2}.$

###### Proof.

Since $H_{1}$ and $H_{2}$ are complex Hilbert spaces, $T_{1}:H_{1}\rightarrow
H_{1}$ and $T_{2}:H_{2}\rightarrow H_{2}$ are self adjoint operators. By the
above proposition, There exists probability measures $\mu_{1},\mu_{2}$ on
$[-\|T_{1}\|,\|T_{1}\|]$ and $[-\|T_{2}\|,\|T_{2}\|]$ and unitary operators
$U_{1}:H_{1}\rightarrow L^{2}(\mu_{1})$ and $U_{2}:H_{2}\rightarrow
L^{2}(\mu_{2})$ such that $U_{1}T_{1}U_{1}^{*}=M_{1}$ and
$U_{2}T_{2}U_{2}^{*}=M_{2}$. $M_{1}$ and $M_{2}$ are canonical multiplication
operators in $L^{2}(\mu_{1})$ and $L^{2}(\mu_{2})$ respectively. Set
$U=e_{1}U_{1}+e_{2}U_{2}$ and $T=e_{1}T_{1}+e_{2}T_{2}$ the idempotent
decomposition. We then have

$U^{*}TU=M=e_{1}M_{1}+e_{2}M_{2}.$

∎

We now prove the spectral theorem for a bounded self adjoint operator.

######  Theorem 3.5.

Let $T$ be a bounded self adjoint operator on a separable bicomplex,
countabely generated Hilbert space $H.$ Then there exists probabilty measures
$\mu_{1},\mu_{2},\ldots,\mu_{1}^{\prime},\mu_{2}^{\prime},\ldots$ on the
intervals $[-\|T_{1}\|,\|T_{1}\|]$ and $[-\|T_{2}\|,\|T_{2}\|]$ and unitary
operators $U_{1}(n):H_{1}\rightarrow L^{2}(\mu_{n})$ and
$U_{2}(n):H_{2}\rightarrow L^{2}(\mu_{n}^{\prime})$ such that

$UTU^{*}=M$

where $M$ is the multiplication operator on
$[e_{1}L^{2}(\mu_{1})+e_{2}L^{2}(\mu_{1}^{\prime})]\oplus[e_{1}L^{2}(\mu_{2})\oplus
e_{2}L^{2}(\mu_{2}^{\prime})]\oplus\ldots$

###### Proof.

Let $\|x_{1}\|_{X}=1$ Let

$S_{1}=\overline{Span\\{x_{1},Tx_{1},T^{2}x_{1},\ldots\\}}$

If $S_{1}=H$ we are done by the above proposition. Else note that $S_{1}$ is
$T$ invariant. Consider $y=T(u)$, $u\in S_{1}$ There is a sequence
$T^{n}x_{1}\rightarrow u$. But $T$ is bounded implies $T^{n+1}x_{1}\rightarrow
Tu$. This means that $y=Tu\in\overline{S_{1}}.$ Also note that
$S_{1}^{\perp}=\\{x\in H:\langle x,y\rangle=0,\text{for all }y\in S^{1}\\}$ is
invariant under $T$. Let $z=T(v)\in T(S_{1}^{\perp})$. For $x\in S^{1},$
$x=\lim T^{n}x_{1}$

$\langle x,z\rangle=\langle\lim T^{n}x_{1},Tv\rangle=\lim\langle
T^{*}T^{n}x_{1},v\rangle=0$

Let $x_{2}\in S_{1}^{\perp}$ and $x_{2}$ be a non null vector $\langle
x_{2}e_{1},x_{2}e_{1}\rangle_{1}\neq 0$(See [3]) and $\langle
x_{2}e_{2},x_{2}e_{2}\rangle_{2}\neq 0.$ So that $\|x_{2}\|_{X}=1.$ and let

$S_{2}=\overline{Span\\{x_{2},Tx_{2},T^{2}x_{2},\ldots\\}}.$

An application of Zorn’s Lemma show that $H$ can be written as a countable
direct sum of $S_{1}.S_{2},\ldots$ in which each $S_{n}$ is a cyclic sub
module and $S_{n}$ are mutually orthogonal.The above proposition can be used
to get the measures $\mu_{n},\mu_{n}^{\prime}$ to get the result.

∎

Acknowledement : The author Akshay S. Rane would like to thank UGC faculty
recharge program, India for their support.

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