# Specialization map for quiver Grassmannians

Giovanni Cerulli Irelli, Francesco Esposito, Xin Fang, Ghislain Fourier  Xin
Fang:
Department Mathematik/Informatik, Universität zu Köln, 50931, Cologne, Germany
<EMAIL_ADDRESS>Ghislain Fourier:
RWTH Aachen University, Chair of Algebra and Representation Theory,
Pontdriesch 10–16, 52062 Aachen, Germany<EMAIL_ADDRESS>
###### Abstract.

We study quiver Grassmannians associated to representations of quivers of type
A. We prove that the specialization map for the integral cohomology associated
to a degeneration of representation is surjective, generalizing a result of
Lanini and Strickland.

##  1\. Introduction

Quiver Grassmannians are natural generalizations of Grassmannians and flag
varieties. Since every projective variety can be realized as a quiver
Grassmannian [17], one has to restrict the study of their topology to those
having representation-theoretic applications, e.g., to canonical bases of
quantum groups, dual canonical bases, and cluster algebras. In [4], a study of
the cohomologies of quiver Grassmannians was initialized by analyzing in
detail the case of quiver Grassmannians attached to indecomposable
representations of the Kronecker quiver. In particular, it was shown that such
quiver Grassmannians admit an affine paving, and their Hilbert polynomials
were explicitly determined. Subsequently, various other classes of quiver
Grassmannians were investigated (see [18], [16], [13], [15], [14], [10], [11])
and some general results were obtained in [5]; namely, all quiver
Grassmannians attached to a quiver of Dynkin type admit affine pavings, thus
their cohomologies are concentrated in even degrees and is generated by
classes attached to algebraic cycles. Furthermore, such properties have also
been proved to hold in almost all cases for quivers of affine type, and the
algebraicity of the cohomology holds in greater generality, i.e. for quiver
Grassmannians attached to rigid representations of acyclic quivers.

Unlike ordinary Grassmannians and flag varieties, quiver Grassmannians vary in
families, the so-called universal quiver Grassmannians. These are projective
families of smooth total space over an affine space, which are equivariant
with respect to the action of a reductive group. The fibers of these maps are
the quiver Grassmannians. Therefore, we may study the variation of the
cohomology of the quiver Grassmannians in these families (see [9] for results
in this direction).

The variation of the cohomology of the fibers of a proper map is encoded in
the constructible sheaves, which are the higher direct images of the constant
sheaf over the total space. These constructible sheaves may be reconstructed
from the specialization maps, which are morphisms from the cohomology of a
fiber to the cohomology of a generization of it. In significant instances of
projective families, e.g. Springer fibers in type $A$ (see [20], [8]), one
observes the surjectivity of the specialization maps, which indicates a strong
form of upper semicontinuity of the cohomology of the fibers of the family in
question.

For (PBW) linear degenerations of flag varieties of type $A$, which were
defined and studied in [3] and [6], the surjectivity of specialization
involving the generic point has been established in [12] using their
realizations as certain Schubert varieties. Linear degenerations of flag
varieties of type $A$ are particular universal quiver Grassmannians for an
equioriented quiver of type $A$. It is hence natural to ask what is the
natural domain of validity of such upper semicontinuity statement for quiver
Grassmannians.

The main result of this paper is the proof of the surjectivity of the
specialization map for any degeneration of representations for a quiver of
type $A$. Such surjectivity of specialization may equivalently be stated as
the surjectivity of the restriction map for the natural embedding of a quiver
Grassmannian of type $A$ in the product of ordinary Grassmannians. One may ask
the question if such surjectivity of specialization holds for quivers of
Dynkin type or for more general quivers.

The proof of the surjectivity of specialization is obtained by reducing to
minimal degenerations, then using a result of Bongartz on minimal
degenerations and applying a suitably generalized version of the reduction
theorem of [5, Theorem 24], to obtain an $\alpha$-partition for which the
specialization maps of the single parts are readily determined. This procedure
thus yields a comparison theorem between the cohomologies of the quiver
Grassmannians of a pair of representations consisting in a minimal
degeneration. This result states that such a specialization map is surjective
and the associated graded of its kernel with respect to a suitable filtration
is described.

###  1.1. Precise statement of the result

Let $\mathcal{Q}$ be a quiver of type $A_{n}$, let $M$, $N$ be two
representations of $\mathcal{Q}$ of the same dimension $\mathbf{d}$ such that
$M$ degenerates to $N$. If $\mathbf{e}$ is any dimension vector for
$\mathcal{Q}$, then there is a specialization map

$c:\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(N),\mathbb{Z})\rightarrow\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$

between the cohomology algebra with integral coefficients of the quiver
Grassmannian $\mathrm{Gr}_{\mathbf{e}}(N)$ of $\mathbf{e}$-dimensional
subrepresentations of $N$ and the quiver Grassmannian of
$\mathbf{e}$-dimensional representations of $M$. It is a homomorphism of
graded algebras.

The result which we prove is the following.

######  Theorem 1.1.

If $\mathcal{Q}$ is a quiver of type $A$, then for any dimension vector
$\mathbf{e}$, the specialization map $c$ is surjective.

This theorem may be stated in the following equivalent form. If
$\mathbf{dim}(M)=\mathbf{d}$, the quiver Grassmannian
$\mathrm{Gr}_{\mathbf{e}}(M)$ is a subvariety of the product
$\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})=\prod_{i\in
Q_{0}}\mathrm{Gr}_{e_{i}}(\mathbb{C}^{d_{i}})$ of ordinary Grassmannians.

######  Theorem 1.2.

The map in cohomology

$\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}}),\mathbb{Z})\rightarrow\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$

induced by the inclusion
$i:\mathrm{Gr}_{\mathbf{e}}(M)\hookrightarrow\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})$,
is a surjective map of commutative graded algebras.

As a corollary we recover the following cohomology surjection results in [12].

Let $\mathcal{Q}$ be an equioriented quiver of type $A_{n}$,
$\mathbf{d}=(n+1,\ldots,n+1)$ and $\mathbf{e}=(1,2,\ldots,n)$ be two dimension
vectors. For $1\leq k\leq n-1$ we fix a tuple
$\mathbf{i}=(i_{1},\ldots,i_{k})\in\mathbb{N}^{k}$ with $1\leq
i_{1}<\ldots<i_{k}\leq n-1$. To this tuple we associate a representation
$M^{\mathbf{i}}$ of $\mathcal{Q}$ defined by:

$M^{\mathbf{i}}=P_{1}^{\oplus
n+1-k}\oplus\left(\bigoplus_{\ell=1}^{k}I_{i_{\ell}}\oplus
P_{i_{\ell}+1}\right),$

where $P_{i}$ (resp. $I_{i}$) are the projective (resp. injective)
representation associated to the vertex $i$. The quiver Grassmannian
$\mathrm{Gr}_{\mathbf{e}}(M^{\mathbf{i}})$ is called a linear degenerate flag
variety of PBW type ([3, Section 5]).

For $M=P_{1}^{\oplus n+1}$, the quiver Grassmannian
$\mathrm{Gr}_{\mathbf{e}}(M)$ is the type $A$ complete flag variety. Since $M$
degenerates to $M^{\mathbf{i}}$, applying Theorem 1.1 to this situation gives:

######  Corollary 1.3 ([12]).

The specialization map
$\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M^{\mathbf{i}}),\mathbb{Z})\rightarrow\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$
is surjective.

Moreover, Theorem 1.1 gives a positive answer to the question in [12, Remark
1]. Acknowledgements: The first author acknowledges support of the University
La Sapienza of Rome (Bando di ateneo RP120172B7D7C329). The second author
acknowledges the support of the project of the University of Padova
BIRD203834/20. The fourth author gratefully acknowledges support by the
Deutsche Forschungsgemeinschaft – Project-ID 286237555– TRR 195.

##  2\. Generalities

In this paper, if not mentioned otherwise, we fix $\mathbb{C}$, the field of
complex numbers, as base field.

###  2.1. Representations of quivers

Let $\mathcal{Q}=(\mathcal{Q}_{0},\mathcal{Q}_{1})$ be a quiver, that is, an
oriented graph; its set of vertices is $\mathcal{Q}_{0}$ and its set of edges
is $\mathcal{Q}_{1}$. We assume furthermore that the quiver $\mathcal{Q}$ is
finite: that is to say, both $\mathcal{Q}_{0}$ and $\mathcal{Q}_{1}$ are
finite sets. A finite-dimensional representation of $\mathcal{Q}$ is the data
$((V_{x})_{x\in\mathcal{Q}_{0}},(f_{\alpha})_{\alpha\in\mathcal{Q}_{1}})$,
where the $V_{x}$ are finite-dimensional vector spaces, and
$f_{\alpha}:V_{s(\alpha)}\rightarrow V_{e(\alpha)}$ are linear maps between
the vector spaces associated with the start and end vertices of the arrow
$\alpha$.

Let $\mathbf{Rep}(\mathcal{Q})$ be the category of finite-dimensional
representations of the quiver $\mathcal{Q}$. It is an abelian category which,
in the case of an acyclic quiver $\mathcal{Q}$, is Krull-Schmidt and
hereditary. Given two $\mathcal{Q}$-representations $M$ and $N$ we denote
$[M,N]=\dim\mathrm{Hom}_{\mathcal{Q}}(M,N)$ and
$[M,N]^{1}=\dim\mathrm{Ext}^{1}_{\mathcal{Q}}(M,N)$.

Let
$V=((V_{x})_{x\in\mathcal{Q}_{0}},(f_{\alpha})_{\alpha\in\mathcal{Q}_{1}})$ be
in $\mathbf{Rep}(\mathcal{Q})$. Its dimension vector is

$(\dim(V_{x}))_{x\in\mathcal{Q}_{0}}\in\mathbb{N}^{\mathcal{Q}_{0}}.$

A subrepresentation of $V$ is the data of a subspace $W_{x}\subseteq V_{x}$
for every vertex $x$ of $\mathcal{Q}$, such that, for every arrow
$\alpha\in\mathcal{Q}_{1}$, one has $f_{\alpha}(W_{s(\alpha)})\subseteq
W_{e(\alpha)}$.

A quiver is of Dynkin type if its underlying unoriented graph is a
disconnected union of simply laced Dynkin diagrams. According to Gabriel’s
theorem, $\mathbf{Rep}(\mathcal{Q})$ has finitely many isoclasses of
indecomposables if and only if $\mathcal{Q}$ is of Dynkin type with finitely
many vertices.

###  2.2. Varieties of representations

Let $\mathbf{d}\in\mathbb{N}^{\mathcal{Q}_{0}}$ be a dimension vector for the
quiver $\mathcal{Q}$ and let $\mathbb{C}^{\mathbf{d}}$ be the
$\mathcal{Q}_{0}$-graded vector space of dimension vector $\mathbf{d}$. The
automorphism group of the graded vector space $\mathbb{C}^{\mathbf{d}}$ is
denoted $\mathrm{G}_{\mathbf{d}}=\mathrm{GL}_{\mathbf{d}}(\mathbb{C})$; it is
a product of general linear groups.

For a Dynkin quiver $\mathcal{Q}$, Gabriel’s theorem implies that there are
only finitely many isoclasses of representations of $\mathcal{Q}$ of dimension
$\mathbf{d}$. Nevertheless, it should not be thought of just as a set, since
one representation may degenerate to another, thus giving rise to a partial
order.

To formalize this, one considers the variety of representations
$\mathrm{R}_{\mathbf{d}}$ which parametrizes isoclasses of pairs
$(V,\mathcal{B})$, where $V$ is a representation of $\mathcal{Q}$ of dimension
vector $\mathbf{d}$, and $\mathcal{B}$ is a homogeneous basis of the graded
vector space $\mathbb{C}^{\mathbf{d}}$. It is an affine algebraic variety
which is a product of matrix spaces. Furthermore, the reductive group
$\mathrm{G}_{\mathbf{d}}$ acts from the right on $\mathrm{R}_{\mathbf{d}}$ by
change of basis. The $\mathrm{G}_{\mathbf{d}}$-orbits in
$\mathrm{R}_{\mathbf{d}}$ correspond to isoclasses of representations of
$\mathcal{Q}$ of dimension $\mathbf{d}$. Furthermore, if $\mathcal{Q}$ is
Dynkin, then there are finitely many $\mathrm{G}_{\mathbf{d}}$-orbits in
$\mathrm{R}_{\mathbf{d}}$. The closure of orbits defines a partial order on
the set of $\mathrm{G}_{\mathbf{d}}$-orbits in $\mathrm{R}_{\mathbf{d}}$; it
is the so-called degeneration order; we denote it $M\leq_{\mathrm{deg}}N$, if
$\mathrm{G}_{\mathbf{d}}\cdot N\subseteq\overline{\mathrm{G}_{\mathbf{d}}\cdot
M}$, and we say that $N$ is a degeneration of $M$.

######  Remark 2.1.

For a Dynkin quiver $\mathcal{Q}$, the degeneration order can be expressed in
terms of the category $\mathbf{Rep}(\mathcal{Q})$ as the $\mathrm{Hom}$ order
or also as the $\mathrm{Ext}$ order. This result is not used directly in the
present paper; however, it is indirectly used through the characterization of
Bongartz of minimal degenerations.

####  2.2.1. Minimal degenerations

We recall the fundamental result of Bongartz on minimal degenerations and
derive from it a consequence used in the study of specialization maps.

######  Theorem 2.2 ((Bongartz, [1], Theorem 4)).

Let $M\leq_{\mathrm{deg}}N$ be a degeneration of representations of a quiver
$\mathcal{Q}$ of Dynkin type. Then, it is minimal if and only if there are
indecomposables $X_{1},S_{1}$, a (nonsplit) short exact sequence

$0\rightarrow X_{1}\rightarrow Y_{1}\rightarrow S_{1}\rightarrow 0$

and $Y^{\prime}$ such that

  1. (i)

$M=Y_{1}\oplus Y^{\prime}$;

  2. (ii)

$N=X_{1}\oplus S_{1}\oplus Y^{\prime}$;

  3. (iii)

for any indecomposable factor $A$ of $N$ that is not isomorphic to $S_{1}$,
one has

$\mathrm{Ext}^{1}(A,M)=\mathrm{Ext}^{1}(A,N);$

  4. (iv)

for any indecomposable factor $B$ of $N$ that is not isomorphic to $X_{1}$,
one has

$\mathrm{Ext}^{1}(M,B)=\mathrm{Ext}^{1}(N,B).$

We now state a consequence of Theorem 2.2 to be used in the study of the
specialization map for type $A$.

######  Theorem 2.3.

Let $M\leq_{\mathrm{deg}}N$ be a minimal degeneration between representations
of a quiver of type $A$. Then, there are

  1. (a)

indecomposables $X_{1}$, $S_{1}$;

  2. (b)

a nonsplit short exact sequence

$\xi_{1}:0\rightarrow X_{1}\rightarrow Y_{1}\rightarrow S_{1}\rightarrow 0;$

  3. (c)

representations $X^{\prime},S^{\prime}$;

satisfying the following conditions:

  1. (i)

$M=Y_{1}\oplus X^{\prime}\oplus S^{\prime}$;

  2. (ii)

$N=X_{1}\oplus S_{1}\oplus X^{\prime}\oplus S^{\prime}$;

  3. (iii)

$\mathrm{Ext}^{1}(S_{1}\oplus S^{\prime},X_{1}\oplus
X^{\prime})=\mathrm{Ext}^{1}(S_{1},X_{1})$;

  4. (iv)

$\mathrm{Ext}^{1}(X_{1},X^{\prime})=\mathrm{Ext}^{1}(S^{\prime},S_{1})=0$.

###### Proof.

From Theorem 2.2, one gets the data $(X_{1},S_{1},\xi_{1},Y^{\prime})$.
Therefore, to define the data $(X_{1},S_{1},\xi_{1},X^{\prime},S^{\prime})$
that meet the requirements of the theorem, one needs to suitably split
$Y^{\prime}$ as a direct sum $X^{\prime}\oplus S^{\prime}$.

Since $Y_{1}$ is the middle term of a nonsplit short exact sequence with
indecomposable extremes, under the type $A$ assumption, it is straightforward
to check that, for any indecomposable $A$, one has either
$\mathrm{Ext}^{1}(A,Y_{1})=0$ or $\mathrm{Ext}^{1}(Y_{1},A)=0$; furthermore,
if $A,B$ are two indecomposables such that $\mathrm{Ext}^{1}(B,Y_{1})\neq 0$
and $\mathrm{Ext}^{1}(Y_{1},A)\neq 0$, then it follows that
$\mathrm{Ext}^{1}(A,B)=0$.

Denote by $\Sigma$ the set of isomorphism classes of indecomposables of
$Y^{\prime}$ that are different from $X_{1}$ and $S_{1}$. Then, the set
$\Sigma^{\prime}=\Sigma\cup\\{Y_{1}\\}$ may be linearly ordered in such a way
that, for $A,B\in\Sigma^{\prime}$ such that $A<B$, one has
$\mathrm{Ext}^{1}(A,B)=0$. With respect to this linear order, let
$Y^{\prime}=X^{\prime}\oplus S^{\prime}$, where $S^{\prime}$ is the sum of the
indecomposables $A$ of $Y^{\prime}$ corresponding to an element of $\Sigma$
for which $A<Y_{1}$ and $X^{\prime}$ is the sum of the indecomposables $B$ of
$Y^{\prime}$ corresponding to an element of $\Sigma$ for which $B>Y_{1}$.

Then, by the property of the ordering, one gets:

$\mathrm{Ext}^{1}(S^{\prime},X^{\prime})=\mathrm{Ext}^{1}(S^{\prime},Y_{1})=\mathrm{Ext}^{1}(Y_{1},X^{\prime})=0.$

Furthermore, since Theorem 2.2 implies that for $A$ an indecomposable of
$X^{\prime}\oplus S^{\prime}$ different from both $X_{1}$ and $S_{1}$, one has

$\mathrm{Ext}^{1}(A,Y_{1})=\mathrm{Ext}^{1}(A,X_{1}\oplus S_{1})\ \
\mathrm{and}\ \ \mathrm{Ext}^{1}(Y_{1},A)=\mathrm{Ext}^{1}(X_{1}\oplus
S_{1},A),$

it follows that

$\mathrm{Ext}^{1}(S^{\prime},X^{\prime})=\mathrm{Ext}^{1}(S^{\prime},S_{1})=\mathrm{Ext}^{1}(X_{1},X^{\prime})=0.$

This concludes the proof. ∎

###  2.3. Generating short exact sequences

Let $\mathcal{Q}$ be a quiver and let

$\textstyle{\xi:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

be a short exact sequence. Following Caldero and Chapoton [2], for every
dimension vector $\mathbf{e}$ there is a map
$\Psi:\mathrm{Gr}_{\mathbf{e}}(Y)\rightarrow\coprod_{\mathbf{f}+\mathbf{g}=\mathbf{e}}\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)$
given by $N\mapsto(\iota^{-1}(N),p(N))$. This is not a nice map in general,
but as noticed in [5], it becomes very useful if $\xi$ has good homological
properties. Following [5], we say that $\xi$ is _generating_ if
$\mathrm{Ext}^{1}(S,X)=\mathbb{C}\xi$. Thus, $\xi$ is either split and
$[S,X]^{1}=0$ or $\xi$ is not split and $[S,X]^{1}=1$. For every
$\mathbf{f}+\mathbf{g}=\mathbf{e}$ we denote
$\mathcal{S}_{\mathbf{f},\mathbf{g}}^{\xi}=\Psi^{-1}(\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S))$
and
$\Psi_{\mathbf{f},\mathbf{g}}^{\xi}:{\mathcal{S}_{\mathbf{f},\mathbf{g}}^{\xi}}\rightarrow\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)$.
We recall ([5, Lemma 20]) that the collection of the strata
$\mathcal{S}_{\mathbf{f},\mathbf{g}}^{\xi}$ forms an $\alpha$-partition of
$\mathrm{Gr}_{\mathbf{e}}(Y)$ in the sense of [8, Section 1.3]. It is now
important to describe the image of $\Psi_{\mathbf{f},\mathbf{g}}^{\xi}$. This
is done in [5, Theorem 31] as follows: to a generating short exact sequence
$\xi$ are associated the two sub-representations

(1) $\begin{array}[]{cc}X_{S}=\mathrm{Ker}(X\rightarrow\tau S)\subset
X,&S^{X}=\mathrm{Im}(\tau^{-}X\rightarrow S)\subset S,\end{array}$

where $\tau$ denotes the AR-translate. Then, [5, Theorem 31] states that
$(N_{1},N_{2})\in\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)$
belongs to the image of $\Psi_{\mathbf{f},\mathbf{g}}^{\xi}$ if and only if
$(N_{1},N_{2})$ does not belong to
$\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\mathbf{dim}S^{X}}(S/S^{X})$.
This means that

(2)
$\mathrm{Im}(\Psi_{\mathbf{f},\mathbf{g}}^{\xi})=\left(\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)\right)\setminus\left(\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\mathbf{dim}S^{X}}(S/S^{X})\right).$

Moreover $\Psi_{\mathbf{f},\mathbf{g}}^{\xi}$ is a (locally trivial) affine
bundle over its image.

We can now interpret Theorem 2.3 in this language and notice that for quivers
of type $A$, every minimal degeneration is given by a generating short exact
sequence.

######  Corollary 2.4.

Let $\mathcal{Q}$ be a quiver of type $A$ and let $M\leq_{\mathrm{deg}}N$ be a
minimal degeneration. Then, in the notation of Theorem 2.3, the short exact
sequence

$\textstyle{\xi:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X=X_{1}\oplus
X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y=M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S=S_{1}\oplus
S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is generating. Moreover, $X_{S}=X^{\prime}\oplus(X_{1})_{S_{1}}$ and
$S^{X}=(S_{1})^{X_{1}}$.

###### Proof.

It follows from Theorem 2.3. ∎

###  2.4. Quiver Grassmannians

Quiver Grassmannians are schematic fibers of projective families called
universal quiver Grassmannians (see [19], [7]) defined as follows: one
considers the incidence variety

$X_{\mathbf{e},\mathbf{d}}=\mathrm{Gr}_{\mathbf{e}}^{Q}(\mathbf{d}):=\\{((U_{i}),(M_{\alpha}))\in\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})\times\mathrm{R}_{\mathbf{d}}\
\ |\ \ M_{\alpha}(U_{s(\alpha)})\subseteq U_{t(\alpha)}\;\forall\alpha\in
Q_{1}\\}$

equipped with two projections

$\textstyle{\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})}$$\textstyle{X_{\mathbf{e},\mathbf{d}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{\mathrm{R}_{\mathbf{d}}}$

which are $\mathrm{G}_{\mathbf{d}}$–equivariant. The map $p_{1}$ realizes
$X_{\mathbf{e},\mathbf{d}}$ as the total space of a homogeneous vector bundle
on $\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})$ (see [7, section 2.2]).
This implies that it is irreducible and smooth.

The map $\pi_{\mathbf{e},\mathbf{d}}=p_{2}$ is proper, and its image is the
closed $\mathrm{G}_{\mathbf{d}}$–stable subvariety of
$\mathrm{R}_{\mathbf{d}}$ consisting of those representations admitting a
subrepresentation of dimension vector $\mathbf{e}$. The morphism
$\pi_{\mathbf{e},\mathbf{d}}$ is the universal family of quiver Grassmannians
of $\mathbf{e}$-dimensional subrepresentations of $\mathbf{d}$-dimensional
representations. For $M\in\mathrm{R}_{\mathbf{d}}$, the quiver Grassmannian
${\mathrm{Gr}_{\mathbf{e}}(M)}$ is the fiber
$\pi_{\mathbf{e},\mathbf{d}}^{-1}(M)$ of $\pi_{\mathbf{e},\mathbf{d}}$ over
$M$.

##  3\. One-dimensional families

Let $0\rightarrow X\rightarrow Y\rightarrow S\rightarrow 0$ be a nonsplit
short exact sequence in $\mathbf{Rep}(\mathcal{Q})$. Fixing bases of $X$ and
$S$ and a section of the map $\pi:Y\rightarrow S$ as graded vector spaces, one
gets two distinct points $M,N\in\mathrm{R}_{\mathbf{d}}$ corresponding
respectively to $Y$ and $X\oplus S$, where $\mathbf{d}=\dim(Y)$. Furthermore,
the $\mathrm{G}_{\mathbf{d}}$-orbit of $M$ closes to the
$\mathrm{G}_{\mathbf{d}}$-orbit of $N$.

Let $\mathcal{L}$ be the affine line in $\mathrm{R}_{\mathbf{d}}$ passing
through $M$ and $N$. Let $T$ be the one-dimensional algebraic sub-torus of
$\mathrm{G}_{\mathbf{d}}$ consisting of those automorphisms of the graded
vector space $X\oplus S$ that act as identity on the subspace $X$ and by
homotheties on the subspace $S$. We define here one-dimensional families of
quiver Grassmannians which play a major role in the comparison of cohomology
of quiver Grassmannians.

######  Definition 3.1.

Let $X_{\mathcal{L}}$ be the inverse image
$\pi_{\mathbf{e},\mathbf{d}}^{-1}(\mathcal{L})$ of $\mathcal{L}$ by the map
$\pi_{\mathbf{e},\mathbf{d}}$ and let
$\pi_{\mathcal{L}}:X_{\mathcal{L}}\rightarrow\mathcal{L}$ be the restriction
of $\pi_{\mathbf{e},\mathbf{d}}$ to $X_{\mathcal{L}}$.

Then, the following statements hold.

######  Proposition 3.2.

  1. (i)

The line $\mathcal{L}$ is $T$-stable. Moreover, identifying $\mathcal{L}$ with
$\mathbb{C}$ through the affine map $\mathbb{C}\rightarrow\mathcal{L}$ which
sends $0,1$ to $N,M$, the $T$-action on $\mathcal{L}$ corresponds to the
natural action of $\mathbb{C}^{\times}$ on $\mathbb{C}$.

  2. (ii)

The fiber of $\pi_{\mathcal{L}}$ over $N$ is $\mathrm{Gr}_{\mathbf{e}}(N)$ and
the family $\pi_{\mathcal{L}}$ is a trivial fibration over
$\mathcal{L}\setminus\\{N\\}$, with fiber $\mathrm{Gr}_{\mathbf{e}}(M)$.

###### Proof.

As mentioned, the space $\mathrm{R}_{\mathbf{d}}$ may be identified with a
product of spaces of matrices, obtained by describing a point of
$\mathrm{R}_{\mathbf{d}}$ by the sequence of matrices associated with the
linear maps of the representation with respect to the fixed basis.

Let us now consider, as above, the points $M,N$ of $\mathrm{R}_{\mathbf{d}}$
corresponding respectively to $Y,X\oplus S$ together with the basis obtained
by the union of the fixed bases of $X$ and $S$, and through the
identification, given by the splitting, of $Y$ and $X\oplus S$ as graded
vector spaces.

With respect to this basis, the linear maps of $N=X\oplus S$ are block-
diagonal

$\begin{pmatrix}f_{\alpha}&0\\\ 0&g_{\alpha}\end{pmatrix}$

and the linear maps of $M=Y$ are block-upper triangular, with block-diagonal
part coinciding with that of $N$

$\begin{pmatrix}f_{\alpha}&n_{\alpha}\\\ 0&g_{\alpha}\end{pmatrix}.$

Thus, the line through $M$ and $N$ in this space of matrices is described by

$t\mapsto\begin{pmatrix}f_{\alpha}&tn_{\alpha}\\\ 0&g_{\alpha}\end{pmatrix},\
\ t\in\mathbb{C}.$

Identify the torus $T$ with $\mathbb{C}^{\times}$, as follows: the complex
number $\lambda\in\mathbb{C}^{\times}$ corresponds to the element of $T$ that
acts by rescaling $S$ by $\lambda$. Then, clearly one has that $\lambda M$
corresponds to the point

$\begin{pmatrix}f_{\alpha}&\lambda n_{\alpha}\\\ 0&g_{\alpha}\end{pmatrix}.$

This proves $(i)$.

The point $(ii)$ follows from the fact that, by definition,
$\pi_{\mathcal{L}}^{-1}(N)=\pi_{\mathbf{e},\mathbf{d}}^{-1}(N)=\mathrm{Gr}_{\mathbf{e}}(N)$,
and the fact that $\pi_{\mathcal{L}}$ is $T$-equivariant and $T$ acts simply
transitively on $\mathcal{L}\setminus\\{N\\}$. ∎

###  3.1. $\alpha$-partitions for one-dimensional families

Let $\xi:0\rightarrow X\rightarrow Y\rightarrow S\rightarrow 0$ be a non-split
generating short exact sequence (see Section 2.3). We define here an
$\alpha$-partition of $X_{\mathcal{L}}$, which extends the $\alpha$-partition
recalled in Section 2.3 on the generic fiber $\mathrm{Gr}_{\mathbf{e}}(M)$.
Moreover, it is $T$-stable and behaves nicely with respect to the map
$\pi_{\mathcal{L}}$.

######  Definition 3.3.

Let $\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}$, with
$\mathbf{f}+\mathbf{g}=\mathbf{e}$ and $i=0,1$, be the subset of
$X_{\mathcal{L}}$ defined as follows:

$\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}=\left\\{(A,B)\in X_{\mathcal{L}}\ \
|\ \ \mathbf{dim}(A\cap X)=\mathbf{f}\ \mathrm{and}\ [A/A\cap X,X/A\cap
X]^{1}=i\right\\}$

######  Lemma 3.4.

The partition
$X_{\mathcal{L}}=\coprod_{\mathbf{f},\mathbf{g},i}\mathcal{S}^{(i)}_{\mathbf{f},\mathbf{g}}$
is an $\alpha$-partition.

###### Proof.

Let us introduce the coarser partition
$X_{\mathcal{L}}=\coprod_{\mathbf{f},\mathbf{g}}\mathcal{S}_{\mathbf{f},\mathbf{g}}$
where
$\mathcal{S}_{\mathbf{f},\mathbf{g}}:=\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(0)}\cup\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(1)}$.
This is a finite partition and for every dimension vector $\mathbf{n}$ the set
$\coprod_{\mathbf{f}+\mathbf{g}\geq\mathbf{n}}\mathcal{S}_{\mathbf{f},\mathbf{g}}$
is closed (here $\geq$ is meant componentwise). It is thus an
$\alpha$-partition. By upper semi-continuity of
$\dim(\mathrm{Ext}^{1}(A/(A\cap X),X/(A\cap X)))$, the partition
$\mathcal{S}_{\mathbf{f},\mathbf{g}}:=\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(0)}\cup\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(1)}$
is an $\alpha$-partition with $\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(1)}$
closed. Combining these two statements together one gets the result. ∎

We consider the following commutative diagram of maps

${\mathcal{S}^{(i)}_{\mathbf{f},\mathbf{g}}}$${X_{\mathcal{L}}}$${\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)\times\mathcal{L}}$${\mathcal{L}}$$\scriptstyle{{j_{\mathbf{f},\mathbf{g}}^{(i)}}}$$\scriptstyle{\Psi_{\mathbf{f},\mathbf{g}}^{(i)}}$$\scriptstyle{\pi_{\mathcal{L}}}$$\scriptstyle{p_{3}}$

where $j_{\mathbf{f},\mathbf{g}}^{(i)}$ is the natural inclusion, the map
$p_{3}$ is the projection on the third factor, and
$\Psi_{\mathbf{f},\mathbf{g}}^{(i)}$ maps $(A,B)$ to $(A\cap X,\pi(A),B)$. We
notice that $\mathcal{S}^{(i)}_{\mathbf{f},\mathbf{g}}$ is
$\mathbb{C}^{\times}$-stable; furthermore, noticing that $\mathbb{C}^{\times}$
acts on
$\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)\times\mathcal{L}$
by its natural action on $\mathcal{L}$, one has that
$\Psi_{\mathbf{f},\mathbf{g}}^{(i)}$ and $p_{3}$ are
$\mathbb{C}^{\times}$-equivariant.

Now we study the parts $\mathcal{S}^{(i)}_{\mathbf{f},\mathbf{g}}$ of the
$\alpha$-partition. Our analysis is based on the techniques of [5, Section 3].

######  Lemma 3.5.

The map $\Psi_{\mathbf{f},\mathbf{g}}^{(i)}$ is a locally trivial affine
bundle on its image. Moreover, the image of
$\Psi_{\mathbf{f},\mathbf{g}}^{(i)}$ is

  1. (i)

$((\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S))-(\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X})))\times\mathcal{L}$,
if $i=0$; and

  2. (ii)

$\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X})\times\\{N\\}$,
if $i=1$.

###### Proof.

Following [5, Section 3], one has two vector bundles $H$ and $K$ on

$\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)\times\mathcal{L}$

and a map $\Phi:H\rightarrow K$ of vector bundles. A splitting as vector
spaces $\theta:S\rightarrow Y$ of the short exact sequence $\xi$ determines a
splitting as vector spaces of all short exact sequences $\xi_{\lambda}$, for
all $\lambda\in\mathbb{C}$. As in [5, Section 3], one gets a section
$z_{\theta}$ of $K$. As in [5, Lemma 22], one shows that there is an
isomorphism of schemes

$\mathcal{S}_{\mathbf{f},\mathbf{g}}=\mathcal{S}^{(0)}_{\mathbf{f},\mathbf{g}}\coprod\mathcal{S}^{(1)}_{\mathbf{f},\mathbf{g}}\cong\Phi^{-1}(z_{\theta}).$

Analogously to [5, Theorem 24], we find that the parts
$\mathcal{S}^{(0)}_{\mathbf{f},\mathbf{g}}$ and
$\mathcal{S}^{(1)}_{\mathbf{f},\mathbf{g}}$ are affine bundles on their images
under $\Phi$. Moreover, as in [5, Theorem 32], one gets the statement (i) of
the lemma.

Statement (ii) follows, using [5, Lemma 31 (6)], by observing that the image
of $\mathcal{S}^{(1)}_{\mathbf{f},\mathbf{g}}$ through $\Phi$ is exactly
$\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}(S)-\dim
S^{X}}(S/S^{X})\times\\{N\\}.$ ∎

######  Lemma 3.6.

Let $\mathcal{Q}$ be a quiver of type $A$ and let $M\leq_{\mathrm{deg}}N$ be a
minimal degeneration associated with the generating short exact sequence
$\textstyle{\xi:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
as in Corollary 2.4. Then both
$\mathrm{Gr}_{\mathbf{f}}(X)\setminus\mathrm{Gr}_{\mathbf{f}}(X_{S})$ and
$\mathrm{Gr}_{\mathbf{g}}(S)\setminus\mathrm{Gr}_{\mathbf{g}\setminus\dim(S^{X})}(S/S^{X})$
admit affine pavings.

###### Proof.

By Corollary 2.4 there are decompositions $X=X_{1}\oplus X^{\prime}$ and
$S=S_{1}\oplus S^{\prime}$ and a generating short exact sequence

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\oplus}$$\textstyle{\oplus}$$\textstyle{\oplus}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}}$$\textstyle{\oplus}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{\prime}}$

Moreover, $X_{S}=X^{\prime}\oplus(X_{1})_{S_{1}}$ and $S^{X}=(S_{1})^{X_{1}}$.
We now prove that
$\mathrm{Gr}_{\mathbf{f}}(X)\setminus\mathrm{Gr}_{\mathbf{f}}(X_{S})$ admits
an affine paving. By Theorem 2.3(iv), the split short exact sequence

$\textstyle{\eta:}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\eta}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{\eta}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is generating split. Thus, we get an $\alpha$-partition
$\mathrm{Gr}_{\mathbf{f}}(X)=\coprod_{\mathbf{f}^{\prime}+\mathbf{f}^{\prime\prime}=\mathbf{f}}\mathcal{S}_{\mathbf{f}^{\prime},\mathbf{f}^{\prime\prime}}^{\eta}$
and vector bundles

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
9.58667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{S}_{\mathbf{f}^{\prime},\mathbf{f}^{\prime\prime}}^{\eta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
33.58667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.58667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Gr}_{\mathbf{f}^{\prime}}(X^{\prime})\times\mathrm{Gr}_{\mathbf{f}^{\prime\prime}}(X_{1})}$}}}}}}}\ignorespaces}}}}\ignorespaces,\
\ N\mapsto(\iota_{\eta}^{-1}(N),p_{\eta}(N)).$

Since $X_{1}$ is indecomposable and $\mathcal{Q}$ is of type $A$,
$\mathrm{Gr}_{\mathbf{f}^{\prime\prime}}(X_{1})$ is either empty or a point.
We notice that $X_{S}=p_{\eta}^{-1}((X_{1})_{S_{1}})$. We conclude that
$\mathrm{Gr}_{\mathbf{f}}(X)\setminus\mathrm{Gr}_{\mathbf{f}}(X_{S})$ is the
union of those strata
$\mathcal{S}_{\mathbf{f}^{\prime},\mathbf{f}^{\prime\prime}}^{\eta}$ such that
$\mathbf{f}^{\prime\prime}\leq\mathbf{dim}(X_{1})_{S_{1}}$. Since
$\mathrm{Gr}_{\mathbf{f}^{\prime}}(X^{\prime})\times\mathrm{Gr}_{\mathbf{f}^{\prime\prime}}(X_{1})$
admits an affine paving by [5], the same holds for all strata
$\mathcal{S}_{\mathbf{f}^{\prime},\mathbf{f}^{\prime\prime}}^{\eta}$ and thus
for $\mathrm{Gr}_{\mathbf{f}}(X)\setminus\mathrm{Gr}_{\mathbf{f}}(X_{S})$.

The proof of the fact that
$\mathrm{Gr}_{\mathbf{g}}(S)\setminus\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X})$
admits an affine paving is obtained using the generating split short exact
sequence $0\rightarrow S_{1}\rightarrow S\rightarrow S^{\prime}\rightarrow 0$
and proceeding analogously. ∎

##  4\. Specialization

In this section, we define the specialization map between the cohomology of
quiver Grassmannians attached to representations related by a degeneration.
Moreover, we study the case of a degeneration attached to a short exact
sequence; in particular, a generating short exact sequence.

We recall here a general sheaf-theoretic result that will be used to define
the specialization map.

######  Proposition 4.1.

Let $Y$ be a complex algebraic variety endowed with a stratification
$\left\\{Y_{\alpha}\right\\}_{\alpha\in A}$, and let $\mathcal{F}$ be a
constructible sheaf on $Y$ whose restriction to any stratum is constant. Let
$y_{\alpha},y_{\beta}$ be points of $Y$, respectively, in the strata
$Y_{\alpha},Y_{\beta}$ satisfying $Y_{\alpha}\subset\overline{Y_{\beta}}$.
Then, there is a well-defined specialization map

$c_{\alpha,\beta}:\mathcal{F}_{y_{\alpha}}\rightarrow\mathcal{F}_{y_{\beta}}$

between the stalk of $\mathcal{F}$ at the point $y_{\alpha}$ and the stalk of
$\mathcal{F}$ at the point $y_{\beta}$. Moreover, for
$y_{\alpha},y_{\beta},y_{\gamma}$ in strata one in the closure of the next,
one has

$c_{\alpha,\gamma}=c_{\beta,\gamma}\circ c_{\alpha,\beta}$

###### Proof.

Since $\mathcal{F}$ is constructible, for every $y\in Y$ there is an open
neighborhood $U$ of $y$ such that $\mathcal{F}_{y}=\mathcal{F}(U)$. Therefore,
one may choose an open neighborhood $U_{\alpha}$ of $y_{\alpha}$ such that
$\mathcal{F}_{y_{\alpha}}=\mathcal{F}(U_{\alpha})$. Furthermore, since
$y_{\alpha}$ is in the closure of the stratum $Y_{\beta}$, it follows that
there is a point $y^{\prime}_{\beta}\in U_{\alpha}\cap Y_{\beta}$. Let
$U_{\beta}$ be a sufficiently small open neighborhood of $y^{\prime}_{\beta}$,
such that $U_{\beta}\subset U_{\alpha}$ and
$\mathcal{F}_{y^{\prime}_{\beta}}=\mathcal{F}(U_{\beta})$. The restriction of
sections then gives a map

$\mathcal{F}_{y_{\alpha}}=\mathcal{F}(U_{\alpha})\rightarrow\mathcal{F}(U_{\beta})=\mathcal{F}_{y^{\prime}_{\beta}}.$

Moreover, since the restriction of $\mathcal{F}$ to the strata of $Y$ is
constant, it follows that there is a canonical isomorphism
$\mathcal{F}_{y^{\prime}_{\beta}}\cong\mathcal{F}_{y_{\beta}}$, and
composition of the previous map with this isomorphism gives the specialization
map

$c_{\alpha,\beta}:\mathcal{F}_{y_{\alpha}}\rightarrow\mathcal{F}_{y_{\beta}}.$

In the case of $y_{\alpha},y_{\beta},y_{\gamma}$ as in the statement of the
proposition, one may find a sufficiently small open neighborhood
$U_{\alpha}\supset U_{\beta}\supset U_{\gamma}$ of
$y_{\alpha},y^{\prime}_{\beta},y^{\prime\prime}_{\gamma}$ respectively, where
$y^{\prime}_{\beta},y^{\prime\prime}_{\gamma}$ are in the strata of
$y_{\beta},y_{\gamma}$ respectively. The restriction of sections from one of
these neighborhoods to another gives the various specialization maps and one
gets the sought equality. ∎

Consider the constant sheaf
$\underline{\mathbb{Z}}_{X_{\mathbf{e},\mathbf{d}}}$ on the universal quiver
Grassmannian $X_{\mathbf{e},\mathbf{d}}$; then the (derived) direct image
$(\pi_{\mathbf{e},\mathbf{d}})_{\ast}\underline{\mathbb{Z}}_{X_{\mathbf{e},\mathbf{d}}}$
is such that its graded cohomology sheaf
$\mathcal{F}=\mathcal{H}((\pi_{\mathbf{e},\mathbf{d}})_{\ast}\underline{\mathbb{Z}}_{X_{\mathbf{e},\mathbf{d}}})$
is a constructible sheaf on $\mathrm{R}_{\mathbf{d}}$. We denote by
$\mathcal{F}_{\mathcal{O}}$ the restriction of $\mathcal{F}$ to a
$\mathrm{G}_{\mathbf{d}}$-orbit $\mathcal{O}$.

######  Lemma 4.2.

For any $\mathrm{G}_{\mathbf{d}}$-orbit $\mathcal{O}$, the local system
$\mathcal{F}_{\mathcal{O}}$ is trivial. The stalk $\mathcal{F}_{M}$ at
$M\in\mathrm{R}_{\mathbf{d}}$ is the cohomology algebra
$\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$.

###### Proof.

Let $M\in\mathcal{O}$. The stabilizer $\textrm{Aut}_{\mathcal{Q}}(M)$ of $M$
in $\mathrm{G}_{\mathbf{d}}$ is connected, being open and dense in the vector
space of endomorphisms of the representation $M$. It follows that the
$\mathrm{G}_{\mathbf{d}}$-equivariant fundamental group of $\mathcal{O}$ is
trivial. Moreover, the local system $\mathcal{F}_{\mathcal{O}}$ is
$\mathrm{G}_{\mathbf{d}}$-equivariant and thus it must be constant. ∎

######  Definition 4.3.

Let $M\leq_{\mathrm{deg}}N$ be a degeneration of representations of
$\mathcal{Q}$ of dimension vector $\mathbf{d}$. Then, the specialization map

$c_{N,M}:\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(N),\mathbb{Z})\rightarrow\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$

is the map obtained by applying Proposition 4.1 to the sheaf
$\mathcal{F}=\mathcal{H}((\pi_{\mathbf{e},\mathbf{d}})_{\ast}\underline{\mathbb{Z}}_{X_{\mathbf{e},\mathbf{d}}})$,
and to the $\mathrm{G}_{\mathbf{d}}$-orbits of $N$ and $M$.

Since the cohomology of quiver Grassmannians of Dynkin type is concentrated in
even degree by [5, Theorem 45], the specialization map is a homomorphism of
commutative graded algebras.

######  Proposition 4.4.

Let $N_{0}=\mathbb{C}^{\mathbf{d}}$ be the semisimple representation of
dimension vector $\mathbf{d}$. Then the specialization map $c_{N_{0},M}$
coincides with the map in cohomology corresponding to the inclusion
$i:\mathrm{Gr}_{\mathbf{e}}(M)\hookrightarrow\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})$.

###### Proof.

By definition, the universal quiver Grassmannian is a subfamily of the
constant family with fiber
$\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})$. Denoting
$\mathcal{F}_{0}=\mathcal{H}(p_{2\ast}(\underline{\mathbb{Z}}_{\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})\times\mathrm{R}_{\mathbf{d}}}))$,
one has a map $\mathcal{F}_{0}\rightarrow\mathcal{F}$, which gives rise to a
commutative diagram

$\begin{CD}\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}}),\mathbb{Z})@>{c_{N_{0},M}}>{}>\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})\\\
@A{\mathrm{id}}A{}A@A{i^{\ast}}A{}A\\\
\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}}),\mathbb{Z})@>{\mathrm{id}}>{}>\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}}),\mathbb{Z})\end{CD}$

where the vertical arrows are the maps on the stalks at $N_{0}$ and $M$,
respectively, corresponding to $\mathcal{F}_{0}\rightarrow\mathcal{F}$, and
the horizontal arrows are specialization maps attached to the sheaves
$\mathcal{F},\mathcal{F}_{0}$ respectively. Thus, we conclude
$c_{N_{0},M}=i^{\ast}$. ∎

######  Proposition 4.5.

Let $\mathcal{Q}$ be a quiver of Dynkin type. The following statements are
equivalent:

  1. (i)

for every minimal degeneration $M\leq_{\mathrm{deg}}N$, the specialization map
$c_{N,M}$ is surjective;

  2. (ii)

for every degeneration $M\leq_{\mathrm{deg}}N$, the specialization map
$c_{N,M}$ is surjective;

  3. (iii)

for every $M\in\mathrm{R}_{\mathbf{d}}$, the map in cohomology
$\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}}),\mathbb{Z})\rightarrow\mathrm{H}^{\ast}(\mathrm{Gr}_{\mathbf{e}}(M),\mathbb{Z})$
induced by the inclusion
$\mathrm{Gr}_{\mathbf{e}}(M)\hookrightarrow\mathrm{Gr}_{\mathbf{e}}(\mathbb{C}^{\mathbf{d}})$
is surjective.

###### Proof.

The equivalence of statements (i) and (ii) follows from the transitivity of
specialization maps established in Proposition 4.1, applied to the sheaf
$\mathcal{H}(\pi_{\mathbf{e},\mathbf{d}})_{\ast}\underline{\mathbb{Z}}_{X_{\mathbf{e},\mathbf{d}}}$.

Furthermore, it follows from Proposition 4.4 that statement (iii) is
equivalent to the surjectivity of specialization maps $c_{N_{0},M}$, where
$N_{0}$ is semisimple. Thus (ii) implies (iii); but one has also, for any
$M\leq_{\mathrm{deg}}N$,

$c_{N_{0},M}=c_{N,M}\circ c_{N_{0},N}$

and hence the surjectivity of $c_{N_{0},M}$ implies the surjectivity of
$c_{N,M}$. This proves the equivalence of (ii) and (iii). ∎

##  5\. Surjectivity of specialization in type $A$

In this section, we study specialization maps in type $A$ and give the proof
of Theorem 1.1 and Theorem 1.2.

We start by studying specializations in the case of minimal degenerations. The
setting is as follows. Let $\mathcal{Q}$ be a quiver of type $A$ and let
$M\leq_{\mathrm{deg}}N$ be a minimal degeneration. By Corollary 2.4, it is
associated with a nonsplit generating short exact sequence, so we may consider
the one-dimensional family
$\pi_{\mathcal{L}}:X_{\mathcal{L}}\rightarrow\mathcal{L}$, following
Definition 3.1. Furthermore, let
$\\{\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}\\}$ be the $\alpha$-partition of
$X_{\mathcal{L}}$ as in Definition 3.3.

Let
$\mathcal{F}=\mathcal{H}(\pi_{\mathcal{L}\ast}\underline{\mathbb{Z}}_{X_{\mathcal{L}}})$
be the cohomology sheaf of the (derived) push-forward of the constant sheaf
$\underline{\mathbb{Z}}_{X_{\mathcal{L}}}$ on $X_{\mathcal{L}}$. Let
$j_{\mathbf{f},\mathbf{g}}^{(i)}$ be the inclusion of the locally closed
subset $\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}$ in $X_{\mathcal{L}}$, and
let $\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$ be the cohomology sheaf
$\mathcal{H}(\pi_{\mathcal{L}\ast}\circ(j_{\mathbf{f},\mathbf{g}}^{(i)})_{!}\
\underline{\mathbb{Z}}_{\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}})$. These
are constructible sheaves of graded algebras on the line $\mathcal{L}$.

######  Proposition 5.1.

The following statements hold.

  * (i)

The graded sheaves $\mathcal{F}$, $\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$
are even, i.e., concentrated in even degree.

  * (ii)

The graded sheaf $\mathcal{F}$ is filtered, the successive subquotients of the
filtration are the sheaves $\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$.

  * (iii)

Such a filtration induces filtrations of $\mathcal{F}_{N}$ and
$\mathcal{F}_{M}$ for which the specialization map is filtered, and the
successive subquotients are the specialization maps
$c_{\mathbf{f},\mathbf{g}}^{(i)}$.

  * (iv)

The specialization map $c_{\mathbf{f},\mathbf{g}}^{(i)}$ is a surjective map;
moreover, it is zero if $i=1$ and it is an isomorphism if $i=0$.

###### Proof.

Let
${p_{3}}:\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S)\times\mathcal{L}\rightarrow\mathcal{L}$
be the projection on the third factor. Being the restriction of the proper map
$\pi_{\mathbf{e},\mathbf{d}}$, $\pi_{\mathcal{L}\ast}=\pi_{\mathcal{L}!}$.
Then one has

$\pi_{\mathcal{L}\ast}\circ(j_{\mathbf{f},\mathbf{g}}^{(i)})_{!}=\pi_{\mathcal{L}!}\circ(j_{\mathbf{f},\mathbf{g}}^{(i)})_{!}=(p_{3})_{!}\circ(\Psi_{\mathbf{f},\mathbf{g}}^{(i)})_{!}$

Furthermore, by Lemma 3.5, the map $\Psi_{\mathbf{f},\mathbf{g}}^{(i)}$ is a
locally trivial affine bundle over its image (which is explicitly described),
it thus follows that $(\Psi_{\mathbf{f},\mathbf{g}}^{(i)})_{!}\
\underline{\mathbb{Z}}_{\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}}$ is
isomorphic to a constant sheaf up to an even shift. By Lemma 3.5, one
concludes that, if $i=0$, the sheaf
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$ is constant on $\mathcal{L}$, with
stalks isomorphic to

${\mathrm{H}_{c}\left((\mathrm{Gr}_{\mathbf{f}}(X)\times\mathrm{Gr}_{\mathbf{g}}(S))\setminus(\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X})),\mathbb{Z}\right)}$

and, if $i=1$, the sheaf $\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$ is
concentrated at $N$, with stalk

$\mathrm{H}_{c}(\mathrm{Gr}_{\mathbf{f}}(X_{S})\times\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X}),\mathbb{Z}).$

In [5, Theorem 45] it was proved that any quiver Grassmannian of Dynkin type
admits an affine paving. This implies that
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(1)}$ is even. Furthermore, using also
Lemma 3.6, it follows that $\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(0)}$ is even
as well. Indeed, we set $C=\mathrm{Gr}_{\mathbf{f}}(X)$,
$C_{1}=\mathrm{Gr}_{\mathbf{f}}(X_{S})$, $D=\mathrm{Gr}_{\mathbf{g}}(S)$,
$D_{1}=\mathrm{Gr}_{\mathbf{g}-\dim(S^{X})}(S/S^{X})$ and get

$(C\times D)\setminus(C_{1}\times D_{1})=[(C\setminus C_{1})\times(D\setminus
D_{1})]\;\coprod\;[C_{1}\times(D\setminus D_{1})]\;\coprod\;[(C\setminus
C_{1})\times D_{1}]\;,$

since $C_{1}$, $D_{1}$, $C\setminus C_{1}$ and $D\setminus D_{1}$ admit affine
pavings ($C_{1}$ and $D_{1}$ by [5, Theorem 45] and $C\setminus C_{1}$ and
$D\setminus D_{1}$ by Lemma 3.6), we conclude that
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(0)}$ is even.

Rearranging and renaming the $\alpha$-partition
$\mathcal{S}_{\mathbf{f},\mathbf{g}}^{(i)}$, one gets a sequence

$\emptyset\subset U_{1}\subset\ldots\subset U_{r-1}\subset
U_{r}=X_{\mathcal{L}}$

of locally closed subsets, such that, for every $i=1,\dots,r$, one has that

$Z_{i}=\bigcup_{j\leq i}U_{i}$

is closed in $X_{\mathcal{L}}$. Thus, a partition

$Z_{i-1}\subset Z_{i}\supset U_{i}$

of $Z_{i}$ into a disjoint union of a closed and an open subset. Denoting
$\eta:Z_{i-1}\rightarrow Z_{i}$ and $\gamma:U_{i}\rightarrow Z_{i}$ the open
and closed inclusions of these subsets into $Z_{i}$, one gets an exact
triangle

$\gamma_{!}\,\underline{\mathbb{Z}}_{U_{i}}\rightarrow\underline{\mathbb{Z}}_{Z_{i}}\rightarrow\eta_{\ast}\,\underline{\mathbb{Z}}_{Z_{i-1}}\rightarrow\cdots$

Applying ${\pi_{\mathcal{L}!}}$ to this exact triangle, one gets an exact
triangle

${\pi_{\mathcal{L}!}\gamma_{!}\,\underline{\mathbb{Z}}_{U_{i}}\rightarrow\pi_{\mathcal{L}!}\,\underline{\mathbb{Z}}_{Z_{i}}\rightarrow\pi_{\mathcal{L}!}\eta_{\ast}\,\underline{\mathbb{Z}}_{Z_{i-1}}\rightarrow\cdots}$

of complexes on $\mathcal{L}$. The graded sheaf of cohomology of the complex
${\pi_{\mathcal{L}!}\gamma_{!}\,\underline{\mathbb{Z}}_{U_{i}}}$ is one of the
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$, thus it is even. By induction,
one may suppose that the cohomology of the complex
${\pi_{\mathcal{L}!}\eta_{\ast}\,\underline{\mathbb{Z}}_{Z_{i-1}}}$ is even.
It follows that the long exact sequence in cohomology attached to the above
exact triangle splits and gives rise to a short exact sequence between
cohomology sheaves.

Since $U_{r}=X_{\mathcal{L}}$, it follows that $\mathcal{F}$ is even. This
completes the proof of $(i)$. Moreover, these short exact sequences give rise
to a filtration of $\mathcal{F}$, the successive quotients of which are the
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$. This proves $(ii)$.

Furthermore, $(iii)$ is just a translation of $(ii)$ in terms of
specialization maps.

Finally, $(iv)$ follows from the fact that the sheaf
$\mathcal{F}_{\mathbf{f},\mathbf{g}}^{(i)}$ is concentrated over $\\{N\\}$ if
$i=1$ and is constant over $\mathcal{L}$ if $i=0$. ∎

###  5.1. Proof of Theorem 1.1 and Theorem 1.2

Let us now give the proof of Theorem 1.1 and Theorem 1.2.

###### Proof.

Observe that Proposition 4.5 implies that Theorem 1.1 and Theorem 1.2 are both
equivalent to proving the surjectivity of the specialization map for the case
of minimal degenerations. Let $M\leq_{\mathrm{deg}}N$ be a minimal
degeneration in type $A$. Then, for $\mathbf{d}$ the dimension vector of $M$
and $N$, the specialization map $c_{N,M}$ is computed by considering the
universal family $\pi_{\mathbf{e},\mathbf{d}}$ and it coincides with the
specialization map attached to the family
$\pi_{\mathcal{L}}:X_{\mathcal{L}}\rightarrow{\mathcal{L}}$, which is the
specialization map associated to the sheaf
$\mathcal{F}=\mathcal{H}({\pi_{\mathcal{L}\ast}}\underline{\mathbb{Z}}_{X_{\mathcal{L}}})$
by Proposition 4.1. Furthermore, by Proposition 5.1, the associated graded of
the specialization map $c_{N,M}$ is surjective. Hence, one concludes that
$c_{N,M}$ is surjective. This concludes the proof. ∎

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# On spectral measures and convergence rates in von Neumann’s Ergodic Theorem

Moacir Aloisio, S. L. Carvalho, C. R. de Oliveira and Edson Souza
Corresponding author.

(September 2022)

###### Abstract

We show that the power-law decay exponents in von Neumann’s Ergodic Theorem
(for discrete systems) are the pointwise scaling exponents of a spectral
measure at the spectral value $1$. We also prove that, under an assumption of
weak convergence, in the absence of a spectral gap, the convergence rates of
the time-average in von Neumann’s Ergodic Theorem depend on sequences of time
going to infinity.

Keywords: von Neumann’s Ergodic Theorem, spectral measures, unitary dynamics.

AMS classification codes: 37A30 (primary), 81Q10 (secondary).

##  1 Introduction

###  1.1 Contextualization and main results

Let $U$ be a unitary operator on a separable complex Hilbert space
$\mathcal{H}$ and let $P^{U}$ be its resolution of the identity. For each
$\psi\in\mathcal{H}$, set

$\psi^{*}:=P^{U}(\\{1\\})\psi.$

Note that $P^{U}(\\{1\\})$ is the orthogonal projection onto the closed
subspace $I(U):=\\{\varphi\mid U\varphi=\varphi\\}$.

Now, recall the following classical result.

######  Theorem 1.1 (von Neumann’s Ergodic Theorem).

Let $U$ be a unitary operator on $\mathcal{H}$. Then, for each
$\psi\in\mathcal{H}$,

$\displaystyle\lim_{K\to\infty}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi=\psi^{*}.$
(1)

The study of convergence rates for ergodic theorems is a long-established
subject in spectral theory and dynamical systems (see [7, 9, 13, 14, 15, 16,
17, 19, 21] and references therein). In the particular case of the von
Neumann’s Ergodic Theorem (vNET), some authors have obtained (power-law)
convergence rates for the time-average in relation (1) in the absence of a
spectral gap at $z=1$ (that is, in case $z=1$ is not an isolated point of the
spectrum of $U$); most of these works are motivated by possible applications
to Koopman operators (see [13, 14, 15, 16, 17] and references therein; see
also [7] for a discussion involving continuous dynamical systems).

On the other hand, if there is a spectral gap at $z=1$, it is well known that
the convergence rates for the time-average in relation (1) are uniform in
$\psi$ and at least of order $K^{-1}$; in what follows, $\sigma(U)$ stands for
the spectrum of $U$.

######  Theorem 1.2 (von Neumann’s Ergodic Theorem with a spectral gap at
$z=1$).

Let $U$ be a unitary operator on $\mathcal{H}$. If there exists
$0<\gamma<\frac{1}{2}$ such that
$\sigma(U)\subset\\{1\\}\cup\\{e^{2i\pi\theta}\mid\theta\in(\gamma,1-\gamma)\\}$,
then for every $K\in\mathbb{N}$,

$\displaystyle\Bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}-P^{U}(\\{1\\})\Bigg{\|}\leq\frac{2}{\gamma\pi
K}.$

The proof of Theorem 1.2 is a consequence of the Spectral Theorem. In order to
put our work into perspective, we recall some details of this proof.

Recall that, by the Spectral Theorem, for every $\varphi\in\mathcal{H}$ there
exists a positive finite Borel measure $\mu_{\varphi}^{U}$ supported on the
circle $\partial\mathbb{D}=\\{z\in\mathbb{C}\mid|z|=1\\}$ such that for any
bounded Borel measurable function $f$ on $\partial\mathbb{D}$, one has

$\|f(U)\varphi\|^{2}=\bigg{\|}\int_{\partial\mathbb{D}}f(z)\,dP^{U}(z)\varphi\bigg{\|}^{2}=\int_{\partial\mathbb{D}}|f(z)|^{2}d\mu_{\varphi}^{U}(z);$

here, $\mu_{\varphi}^{U}$ stands for the so-called spectral measure of $U$
with respect to $\varphi$.

We add another ingredient needed in the proof of Theorem 1.2.

######  Lemma 1.1 (von Neumann’s Lemma).

Let $U$ be a unitary operator on $\mathcal{H}$. Then, for each
$\psi\in\mathcal{H}$ and each $K\in\mathbb{N}$,

$\displaystyle\Bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\Bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{\partial\mathbb{D}}\left|\frac{z^{K}-1}{z-1}\right|^{2}d\mu_{\psi-\psi^{*}}^{U}(z).$
(2)

######  Remark 1.1.

Relation (2) indicates that the convergence rates of the time-average in (1)
depend only on local scale properties of $\mu_{\psi-\psi^{*}}^{U}$ at $z=1$.
In this context, this relation arises from an argument due to von Neumann,
presented in [21] to prove the Quasi-Ergodic Hypothesis, and it was used more
recently by Kachurovskii and Sedalishchev [14] to relate the local scale
properties of $\mu_{\psi-\psi^{*}}^{U}$ at $z=1$ with the power-law
convergence rates in (1) (see Theorem 1.3 ahead).

###### Proof (Lemma 1.1).

For each $K\in\mathbb{N}$ and each $z\in\partial\mathbb{D}\setminus\\{1\\}$,
recall that

$\displaystyle\displaystyle\sum_{j=0}^{K-1}z^{j}$ $\displaystyle=$
$\displaystyle\frac{z^{K}-1}{z-1}.$

Note that, for each $\psi\in\mathcal{H}$,
$\mu_{\psi-\psi^{*}}^{U}(\\{1\\})=0$. Thus, by the Spectral Theorem, it
follows that for each $\psi\in\mathcal{H}$ and each $K\in\mathbb{N}$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}(\psi-\psi^{*})\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}\int_{\partial\mathbb{D}}z^{j}dP^{U}(z)(\psi-\psi^{*})\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\bigg{\|}{\frac{1}{K}}\int_{\partial\mathbb{D}}\frac{z^{K}-1}{z-1}dP^{U}(z)(\psi-\psi^{*})\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{\partial\mathbb{D}}\bigg{|}\frac{z^{K}-1}{z-1}\bigg{|}^{2}d\mu_{\psi-\psi^{*}}^{U}(z).$

∎

###### Proof (Theorem 1.2).

Since, for every $\psi\in\mathcal{H}$,
supp$(\mu_{\psi-\psi^{*}}^{U})\subset\sigma(U)\subset\\{1\\}\cup\\{e^{2i\pi\theta}\mid\theta\in(\gamma,1-\gamma)\\}$
and $\mu_{\psi-\psi^{*}}^{U}(\\{1\\})=0$, it follows from Lemma 1.1 that for
each $\psi\in\mathcal{H}$ and each $K\in\mathbb{N}$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{\partial\mathbb{D}}\left|\frac{z^{K}-1}{z-1}\right|^{2}d\mu_{\psi-\psi^{*}}^{U}(z)$
$\displaystyle\leq$
$\displaystyle\frac{1}{K^{2}}\left(\sup_{z\in\\{e^{2i\pi\theta}\mid\theta\in(\gamma,1-\gamma)\\}}\left|\frac{z^{K}-1}{z-1}\right|^{2}\right)\mu_{\psi-\psi^{*}}^{U}(\partial\mathbb{D})$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\left(\sup_{\theta\in(\gamma,1-\gamma)}\left|\frac{\sin(K\pi\theta)}{\sin(\pi\theta)}\right|^{2}\right)\|\psi-\psi^{*}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{1}{(K\sin(\pi\gamma))^{2}}\,\|(I-P^{U}({\\{1\\}}))\psi\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4}{\gamma^{2}\pi^{2}K^{2}}\,\|\psi\|^{2},$

where we have used that $\gamma\pi/2\leq\sin(\pi\gamma)$ for each
$0<\gamma<1/2$. Hence, for every $K\in\mathbb{N}$,

$\displaystyle\Bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}-P^{U}(\\{1\\})\Bigg{\|}\leq\frac{2}{\gamma\pi
K}.$

∎

Next, we recall the result due to Kachurovskii and Sedalishchev [14],
mentioned in Remark 1.1, that relates the (power-law) convergence rates in
vNET with the local scale properties of $\mu_{\psi-\psi^{*}}^{U}$ at $z=1$, in
case there is no spectral gap.

In what follows, for each $0<\epsilon<\frac{1}{2}$, set

$S_{\epsilon}:=\big{\\{}e^{2\pi
i\theta}\mid-\epsilon<\theta<\epsilon\big{\\}}.$

######  Theorem 1.3 (Theorem 1 in [14]).

Let $U$ be a unitary operator on $\mathcal{H}$, $\psi\in\mathcal{H}$ and
$0<\alpha<2$.

  1. i)

If there exists $C_{\psi}>0$ so that, for every $0<\epsilon<\frac{1}{2}$,
$\mu_{\psi-\psi^{*}}^{U}(S_{\epsilon})\leq C_{\psi}\epsilon^{\alpha}$, then
for every $K\in\mathbb{N}$ there exists ${\widetilde{C}_{\psi}}>0$ such that

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}\leq\frac{{\widetilde{C}_{\psi}}}{K^{\alpha}}\,.$

  2. ii)

Conversely, if there exists $C_{\psi}>0$ such that for every $K\in\mathbb{N}$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}\leq\frac{{C_{\psi}}}{K^{\alpha}},$

then there exists ${\widetilde{C}_{\psi}}>0$ so that for every
$0<\epsilon<\frac{1}{2}$,
$\mu_{\psi-\psi^{*}}^{U}(S_{\epsilon})\leq{\widetilde{C}_{\psi}}\epsilon^{\alpha}$.

######  Remark 1.2.

Although Theorem 1.3 was originally stated for Koopman operators, its proof
holds for any unitary operator.

It is clear from Lemma 1.1 and Theorem 1.3 that, in this setting, the behavior
of the power-law convergence rates in vNET depends only on the local scale
properties of $\mu_{\psi-\psi^{*}}^{U}$ at $z=1$. Taking this into account,
our first result in this work shows (under some mild assumptions) that these
convergence rates are indeed ruled by the lower and upper pointwise exponents
of $\mu_{\psi-\psi^{*}}^{U}$ at $z=1$.

Recall that the lower and upper pointwise exponents of a finite (positive)
Borel measure on $\partial\mathbb{D}$, say $\mu$, at $z=1$ are defined,
respectively, by

$d_{\mu}^{-}(1):=\liminf_{\epsilon\downarrow
0}\frac{\ln\mu(S_{\epsilon})}{\ln\epsilon}\quad{\rm and}\quad
d_{\mu}^{+}(1):=\limsup_{\epsilon\downarrow
0}\frac{\ln\mu(S_{\epsilon})}{\ln\epsilon}\,,$

if, for all small enough $\epsilon>0$, $\mu(S_{\epsilon})>0$; one sets
$d_{\mu}^{\mp}(w):=\infty$, otherwise.

######  Theorem 1.4.

Let $U$ be a unitary operator on $\mathcal{H}$, $0\neq\psi\in\mathcal{H}$ and
suppose that $0\leq d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)\leq 2$. Then,

$-\liminf_{K\to\infty}\frac{\ln\bigg{\|}{\dfrac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}}{\ln
K}=d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1),$ (3)
$-\limsup_{K\to\infty}\frac{\ln\bigg{\|}{\dfrac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}}{\ln
K}=d^{-}_{\mu_{\psi-\psi^{*}}^{U}}(1).$ (4)

######  Remark 1.3.

It is well known that for Koopman operators, for every $\psi\in\mathcal{H}$
such that $\psi-\psi^{*}\not=0$, the respective time-average presented in vNET
does not converge to $\psi^{*}$ faster than $K^{-2}$ (see, for instance,
Corollary 5 in [11]). In this sense, since most of the possible applications
of Theorem 1.4 are related with Koopman operators, and for these operators
$0\leq d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)\leq 2$, the hypothesis that $0\leq
d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)\leq 2$ is quite reasonable.

######  Remark 1.4.

It is worth underlying that the problem of obtaining the actual values of the
decaying exponents of time-averages in terms of the dimensional properties of
measures in dynamical systems and quantum dynamics is very natural and
recurring in the literature; see, for instance, [2, 4, 5, 6, 9, 12]. In this
context, Theorem 1.4 does add something to the body of knowledge on power-law
convergence rates in vNET; namely, it establishes an explicit relation between
the power-law convergence rates of such time-average and the local scale
spectral properties of the unitary operator $U$ at $z=1$.

By Theorem 1.4, if for some $\psi\in\mathcal{H}$ one has

$0\leq
d^{-}_{\mu_{\psi-\psi^{*}}^{U}}(1)<d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)\leq 2,$
(5)

then the power-law convergence rates (related to $\psi$) in vNET actually
depend on sequences of time going to infinity.

Write $L:=UP^{U}(\partial\mathbb{D}\setminus\\{1\\})$; here, we also prove
that the condition (5) is satisfied for a unitary operator $U$ and for a
$G_{\delta}$ dense set of vectors $\psi\in\mathcal{H}$, if $L^{j}$ converges
weakly to zero (i.e., for each pair $\psi,\varphi\in\mathcal{H}$, $\langle
L^{j}\psi,\varphi\rangle\to 0$ as $j\to\infty$) and if $1$ is an accumulation
point of $\sigma(U)$.

######  Theorem 1.5.

Let $U$ be a unitary operator on $\mathcal{H}$ such that $1$ is an
accumulation point of its spectrum $\sigma(U)$. If $L^{j}$ (recall that
$L=UP^{U}(\partial\mathbb{D}\setminus\\{1\\})$) converges to zero in the weak
operator topology, then the set of $\psi\in\mathcal{H}$ such that, for all
$\epsilon>0$,

$\displaystyle\limsup_{K\to\infty}K^{\epsilon}\,\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}=\infty$

and

$\displaystyle\liminf_{K\to\infty}K^{2-\epsilon}\,\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}=0,$

is a dense $G_{\delta}$ set in $\mathcal{H}$.

######  Remark 1.5.

The results presented in this work are analogous to those obtained by the
first three authors in [2] on the convergence rates of normal semigroups (see
also [1, 8, 9]). The main technical difference is that, for normal semigroups,
it is enough to analyze the (self-adjoint) real part of the generator in order
to evaluate the power-law convergence rates, whereas here one needs to analyze
spectral properties of unitary operators.

We note that an adaptation of the arguments in [2] could be made in order to
study the convergence rates in vNET for continuous dynamical systems, since
this case leads naturally to self-adjoint generators.

###  1.2 Organization of the text

The organization of this work is as follows. In Section 2, we illustrate our
general results by presenting an explicit application of Theorem 1.5 to
Koopman operators. The proof of Theorems 1.4 and 1.5 are left to Section 3.

Some words about notation: $U$ will always denote a unitary operator acting on
the separable complex Hilbert space $\mathcal{H}$; we denote its spectrum by
$\sigma(U)$, and by $\mu_{\varphi}^{U}$ the spectral measure of $U$ associated
with $\varphi\in\mathcal{H}$;
$\partial\mathbb{D}=\\{z\in\mathbb{C}\mid|z|=1\\}$; for each Borel set
$\Lambda\subset\partial\mathbb{D}$, $P^{U}(\Lambda)$ represents the resolution
of the identity of $U$ at $\Lambda$.

##  2 Application to Koopman operators

Let $(X,\mathcal{A},m)$ be a Lebesgue space with continuous measure $m$ (i.e.,
a non-atomic probability space obtained by the completion of a Borel measure
on a complete separable metric space); for every automorphism $T$ on $X$, one
defines the Koopman operator related with $T$ by the law

$U_{T}:L_{m}^{2}(X)\to L_{m}^{2}(X),\quad(U_{T}f)(x):=f(Tx),\quad x\in X.$

We say that $\sigma(U_{T})$ is purely absolutely continuous on
$\partial\mathbb{D}\setminus\\{1\\}$ if $U_{T}$ is purely absolutely
continuous on

$\big{\\{}f\in L_{m}^{2}(X)\mid\textstyle\int_{X}f(x)\,dm(x)=0\big{\\}};$

observe that $1$ is always an eigenvalue of the Koopman operator, since
$U_{T}\varphi=\varphi$ for every constant function $\varphi$.

######  Theorem 2.1.

Let $T$ be an automorphism on $X$ with purely absolutely continuous spectrum
on $\partial\mathbb{D}\setminus\\{1\\}$ such that $1$ is an accumulation point
of $\sigma(U_{T})$. Then, the set of vectors $\psi\in L_{m}^{2}(X)$ so that,
for all $\epsilon>0$,

$\displaystyle\limsup_{K\to\infty}K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U_{T}^{j}\psi-\psi^{*}\bigg{\|}^{2}=\infty$

and

$\displaystyle\liminf_{K\to\infty}K^{2-\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U_{T}^{j}\psi-\psi^{*}\bigg{\|}^{2}=0,$

is a dense $G_{\delta}$ set in $L_{m}^{2}(X)$.

######  Remark 2.1.

  1. i)

The result stated in Theorem 2.1 applies to $K$-automorphisms, in particular
to Bernoulli shifts [20].

  2. ii)

Note that under the assumptions of Theorem 2.1, it follows from Theorem 1.4
and Remark 1.3 that for a typical $\psi\in L_{m}^{2}(X)$,
$d^{-}_{\mu_{\psi-\psi^{*}}^{T}}(1)=0$ and
$d^{+}_{\mu_{\psi-\psi^{*}}^{T}}(1)=2$.

The proof of Theorem 2.1 is a direct consequence of Theorem 1.5 and of the
next result.

######  Proposition 2.1.

Let $U$ be a unitary operator on $\mathcal{H}$ with absolutely continuous
spectrum in $\partial\mathbb{D}\setminus\\{1\\}$ and
$L=UP^{U}(\partial\mathbb{D}\setminus\\{1\\})$. Then, $L^{j}$ converges to
zero in the weak operator topology as $j\to\infty$.

###### Proof.

For each $\phi\in\mathcal{H}$, let

$\mathcal{H}_{\phi}:=\big{\\{}P^{U}(g)\phi\,\mid\,g\in
L^{2}_{\mu_{\phi}^{U}}(\partial\mathbb{D})\big{\\}}$

be the cyclic subspace generated by $\phi$, $P_{\phi}$ be the corresponding
orthogonal projection onto $\mathcal{H}_{\phi}$ and $F:\mathcal{H}_{\phi}\to
L^{2}_{\mu_{\phi}^{U}}(\partial\mathbb{D})$ be the unitary operator given by
the law $F(P^{U}(g)\phi)=g$ (note that
$L=P^{U}(g\chi_{\partial\mathbb{D}\setminus\\{1\\}})$, with $g(z)=z$).

Fix $\psi,\varphi\in\mathcal{H}$ and let $h\in
L^{2}_{\mu_{\psi}^{U}}(\partial\mathbb{D})$ be such that
$P_{\psi}\varphi=P^{U}(h)\psi$. Let also $\\{g_{n}\\}$ be a sequence in the
space of bounded Borel functions in $\partial\mathbb{D}$, which is dense in
$L^{2}_{\mu_{\psi}^{U}}(\partial\mathbb{D})$, with
$||g_{n}-h||_{L^{2}_{\mu_{\psi}^{U}}(\partial\mathbb{D})}\to 0$ as
$n\to\infty$.

Now, since $\mu_{\psi}^{U}(\cdot\cap\partial\mathbb{D}\setminus\\{1\\})$ is
absolutely continuous, it follows from Radon-Nikodym Theorem that there exists
$f\in L^{1}(\partial\mathbb{D}\setminus\\{1\\})$ such that
$d\mu_{\psi}^{U}(z)=f(z)dz$. Then, $g_{n}f\in
L^{1}(\partial\mathbb{D}\setminus\\{1\\})$ for each $n\in\mathbb{N}$, and so,
it follows from Riemann-Lebesgue Lemma that for every $n\in\mathbb{N}$,

$\displaystyle\lim_{j\to\infty}\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{j\to\infty}\int_{\partial\mathbb{D}\setminus\\{1\\}}z^{j}g_{n}(z)d\mu_{\psi}^{U}(z)$
(6) $\displaystyle=$
$\displaystyle\lim_{j\to\infty}\int_{\partial\mathbb{D}\setminus\\{1\\}}z^{j}g_{n}(z)f(z)dz$
$\displaystyle=$ $\displaystyle\lim_{j\to\infty}\int_{0}^{1}e^{2i\pi\theta
j}g_{n}(e^{2i\pi\theta})f(e^{2i\pi\theta})d\theta=0;$

we have used the fact that the Lebesgue measure over $\partial\mathbb{D}$ is
the pushforward, with respect to the map
$h:[0,1)\rightarrow\partial\mathbb{D}$, $h(\theta)=e^{2\pi i\theta}$, of the
Lebesgue measure over $[0,1)$.

On the other hand, for each $j\geq 0$, one has

$\displaystyle\big{|}\big{\langle}L^{j}\psi,\varphi\big{\rangle}-\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\big{\langle}P_{\psi}L^{j}\psi,\varphi\big{\rangle}-\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\big{\langle}L^{j}\psi,P_{\psi}\varphi\big{\rangle}-\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}\big{\langle}L^{j}\psi,P^{U}(h)\psi-P^{U}(g_{n})\psi\big{\rangle}\big{|}$
$\displaystyle\leq$ $\displaystyle\|\psi\|\,\|P^{U}(h)\psi-P^{U}(g_{n})\psi\|$
$\displaystyle=$
$\displaystyle\|\psi\|\,\|h-g_{n}\|_{L^{2}_{\mu_{\psi}^{U}}(\partial\mathbb{D})},$

and so

$\displaystyle\lim_{n\to\infty}\big{|}\big{\langle}L^{j}\psi,\varphi\big{\rangle}-\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}\big{|}=0;$

note that this convergence is uniform on $j$. Therefore, by combining this
result with relation (6) and Moore-Osgood Theorem, one gets

$\displaystyle\lim_{j\to\infty}\big{\langle}L^{j}\psi,\varphi\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{j\to\infty}\lim_{n\to\infty}\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\lim_{j\to\infty}\big{\langle}L^{j}\psi,P^{U}(g_{n})\psi\big{\rangle}=0.$

∎

##  3 Proofs of Theorems 1.4 and 1.5

###  3.1 Proof of Theorem 1.4

An important ingredient in the proof of Theorem 1.4 is Lemma 3.1, which may be
obtained by using essentially the same arguments presented in the proof of
Lemma 3.2 in [10]; we, therefore, omit its proof.

######  Lemma 3.1.

Let $U$ be a unitary operator on $\mathcal{H}$ and $0\neq\psi\in\mathcal{H}$.
Let also $(K_{l})$ be a sequence of positive integers satisfying
$K_{l}\to\infty$ as $l\to\infty$. If for some $0<\alpha<2$ there exists
$C_{\psi}>0$ so that, for every $K_{l}$, $\mu_{\psi-\psi^{*}}(S_{1/K_{l}})\leq
C_{\psi}K_{l}^{-\alpha}$, then there exists ${\widetilde{C}_{\psi}}>0$ such
that for every $K_{l}$,

$\displaystyle\int_{\partial\mathbb{D}}\left|\frac{z^{K_{l}}-1}{z-1}\right|^{2}d\mu_{\psi-\psi^{*}}^{U}(z)\leq{\widetilde{C}_{\psi}}\,K_{l}^{2-\alpha}.$

Let $f:[0,1)\rightarrow\partial\mathbb{D}$ denote the map $f(\theta)=e^{2\pi
i\theta}$ and let $f^{*}\mu$ be the pull-back, with respect to $f$, of the
positive Borel measure $\mu$ on $\partial\mathbb{D}$. It follows from Lemma
1.1 that for each $K\in\mathbb{N}$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{\partial\mathbb{D}}\left|\frac{z^{K}-1}{z-1}\right|^{2}d\mu_{\psi-\psi^{*}}^{U}(z)$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{0}^{1}\left|\frac{e^{2i\pi\theta
K}-1}{e^{2i\pi\theta}-1}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{0}^{1}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta).$

Given that for every $0<\epsilon<1$,

$\int_{1-\epsilon}^{1}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta)=\int_{-\epsilon}^{0}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta),$

it follows that

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{K^{2}}\int_{-\epsilon}^{\epsilon}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta)$
$\displaystyle+$
$\displaystyle\frac{1}{K^{2}}\int_{\epsilon}^{1-\epsilon}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta)$
$\displaystyle\geq$
$\displaystyle\frac{1}{K^{2}}\int_{-\epsilon}^{\epsilon}\left|\frac{\sin(\pi\theta
K)}{\sin(\pi\theta)}\right|^{2}d(f^{*}\mu_{\psi-\psi^{*}}^{U})(\theta).$

Since for every $-\pi/2\leq\theta\leq\pi/2$,
$\displaystyle\frac{|\theta|}{2}\leq|\sin(\theta)|\leq 2|\theta|$, it follows
that for each $K\geq 2$ and each $0<\epsilon<1/K$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}\geq\frac{(f^{*}\mu_{\psi-\psi^{*}}^{U})(-\epsilon,\epsilon)}{16}=\frac{\mu_{\psi-\psi^{*}}^{U}(S_{\epsilon})}{16}.$

Hence,

$-\liminf_{K\to\infty}\frac{\ln\bigg{\|}{\dfrac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}}{\ln
K}\leq d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1),$
$-\limsup_{K\to\infty}\frac{\ln\bigg{\|}{\dfrac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}}{\ln
K}\leq d^{-}_{\mu_{\psi-\psi^{*}}^{U}}(1).$

Now, if $d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)=0$, then the complementary
inequality in (3) follows readily; otherwise, for each
$0<\epsilon<d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)\leq 2$, there exists a sequence
of positive integers $(K_{l})$ with
$\displaystyle\lim_{l\to\infty}K_{l}=\infty$ such that for each
$l\in\mathbb{N}$,

$\mu_{\psi-\psi^{*}}^{U}(S_{1/K_{l}})\leq
K_{l}^{\epsilon-d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)}.$

So, by Lemmas 1.1 and 3.1, it follows that there exists $C_{\psi,\epsilon}>0$
such that

$\displaystyle\bigg{\|}{\frac{1}{K_{l}}}\displaystyle\sum_{j=0}^{K_{l}-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
C_{\psi,\epsilon}K_{l}^{\epsilon-d^{+}_{\mu_{\psi-\psi^{*}}}(1)}.$

Thus,

$-\liminf_{K\to\infty}\frac{\ln\bigg{\|}{\dfrac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}}{\ln
K}\geq d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)-\epsilon.$

Given that $0<\epsilon<d^{+}_{\mu_{\psi-\psi^{*}}^{U}}(1)$ is arbitrary, the
complementary inequality in (3) follows.

The proof of the complementary inequality in (4) is analogous, and so it is
omitted. ∎

###  3.2 Proof of Theorem 1.5

We begin with some preparation.

######  Theorem 3.1 (Theorem 1 in [3]).

Let $L$ be a bounded linear operator on $\mathcal{H}$ such that $L^{j}$
converges to zero in the weak operator topology. Suppose that $1\in\sigma(L)$.
Then, for each sequence $(a_{j})_{j\geq 0}$ of positive numbers satisfying
$\displaystyle\lim_{j\to\infty}a_{j}=0$ and for each $\delta>0$, there exists
$\eta\in\mathcal{H}$ such that
$\|\eta\|<\displaystyle\sup_{j}\\{a_{j}\\}+\delta$ and

${\rm Re}\,\langle L^{j}\eta,\eta\rangle>a_{j},\quad\forall j\geq 0.$

######  Lemma 3.2.

Let $U$ be a unitary operator on $\mathcal{H}$,
$L=UP^{U}(\partial\mathbb{D}\setminus\\{1\\})$, and suppose that $L^{j}$
converges to zero in the weak operator topology. If $1$ is an accumulation
point of $\sigma(U)$, then for each $0<\epsilon<1$, there exists
$\eta\in\mathcal{H}$ with $\|\eta\|\leq 1$ so that

$\displaystyle\liminf_{K\to\infty}K^{\epsilon}\,\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}=\infty.$

###### Proof.

Observe that:

  1. i)

$1\in\sigma(L)$, since $1$ is an accumulation point of $\sigma(U)$.

  2. ii)

$L^{j}$ converges to zero in the weak operator topology.

  3. iii)

By the spectral functional calculus,
$UP^{U}(\partial\mathbb{D}\setminus\\{1\\})=P^{U}(\partial\mathbb{D}\setminus\\{1\\})U$,
and so, for $j\in\mathbb{N}$,

$\displaystyle L^{j}$ $\displaystyle=$ $\displaystyle U^{j}(1-P^{U}(\\{1\\}))$
$\displaystyle=$ $\displaystyle
U^{j}-U^{j}P^{U}(\\{1\\})=U^{j}-P^{U}(\\{1\\}).$

Let $0<\epsilon^{\prime}<1$ be such that $0<2\epsilon^{\prime}<\epsilon<1$. By
Theorem 3.1, there exists $\eta\in\mathcal{H}$ with $\|\eta\|\leq 1$ such
that, for each $j\geq 0$,

${\rm Re}\,\langle L^{j}\eta,\eta\rangle>\frac{1}{(j+2)^{\epsilon^{\prime}}};$

thus, for each $K>K_{0}:=2^{(2-\epsilon^{\prime})/(1-\epsilon^{\prime})}$,

$\displaystyle{\rm
Re}\,\biggr{\langle}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}L^{j}\eta,\eta\biggr{\rangle}$
$\displaystyle>$
$\displaystyle{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}\frac{1}{(j+2)^{\epsilon^{\prime}}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{K}\int_{0}^{K-1}\dfrac{1}{(x+2)^{\epsilon^{\prime}}}dx$
$\displaystyle>$
$\displaystyle\frac{K^{1-\epsilon^{\prime}}-2^{1-\epsilon^{\prime}}}{K(1-\epsilon^{\prime})}>\frac{C}{K^{\epsilon^{\prime}}},$

with $C=1/(2(1-\epsilon^{\prime}))$. By iii) above, for $K>K_{0}$,

$\displaystyle{\rm
Re}\,\biggr{\langle}{\frac{1}{K}}\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*},\eta\biggr{\rangle}$
$\displaystyle=$ $\displaystyle{\rm
Re}\left(\biggr{\langle}{\frac{1}{K}}\sum_{j=1}^{K-1}L^{j}\eta,\eta\biggr{\rangle}+\biggr{\langle}{\frac{1}{K}}(\eta-\eta^{*}),\eta\biggr{\rangle}\right)$
$\displaystyle>$ $\displaystyle\frac{C}{K^{\epsilon^{\prime}}}-{\rm
Re}\,\biggr{\langle}{\frac{\eta^{*}}{K}},\eta\biggr{\rangle}.$

By Cauchy-Schwarz Inequality, for $K>K_{0}$,

$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}>\frac{C}{K^{\epsilon^{\prime}}}-\frac{1}{K}$
$\displaystyle\Rightarrow$
$\displaystyle\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}>\frac{C^{2}}{K^{2\epsilon^{\prime}}}-\frac{2C}{K^{1+\epsilon^{\prime}}}+\frac{1}{K^{2}}.$

Hence,

$\displaystyle\liminf_{K\to\infty}K^{\epsilon}\,\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}=\infty,$

and we are done. ∎

Let us proceed to the proof of Theorem 1.5. Let $\epsilon>0$; since for each
$K\in\mathbb{N}$, the map

${\mathcal{H}}\ni\psi\;\longmapsto
K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}$

is continuous, it follows that

$\displaystyle G^{+}(\epsilon)$ $\displaystyle:=$
$\displaystyle\biggl{\\{}\psi\mid\limsup_{K\to\infty}K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}=\infty\biggl{\\}}$
$\displaystyle=$ $\displaystyle\bigcap_{n\geq 1}\bigcap_{l\geq
1}\bigcup_{K\geq l}\biggl{\\{}\psi\mid
K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}>n\biggl{\\}}$

is a $G_{\delta}$ set in ${\mathcal{H}}$. The proof that

$\displaystyle
G^{-}(\epsilon):=\biggl{\\{}\psi\mid\liminf_{K\to\infty}K^{2-\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}=0\biggl{\\}}$

is also a $G_{\delta}$ set in ${\mathcal{H}}$ is completely analogous, and so
we omit it.

Claim: For each $\epsilon>0$, $G^{+}(\epsilon)$ and $G^{-}(\epsilon)$ are
dense sets in $\mathcal{H}$.

It follows from Claim and Baire Category Theorem that

$\mathcal{G}:=\bigcap_{\epsilon\in\mathbb{Q}\cap(0,\infty)}\biggr{(}G^{+}(\epsilon)\cap
G^{-}(\epsilon)\biggr{)}$

is also a dense $G_{\delta}$ set in ${\mathcal{H}}$. Since for each
$\psi\in\mathcal{G}$ and each $\epsilon>0$,

$\displaystyle\limsup_{K\to\infty}K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}=\infty$

and

$\displaystyle\liminf_{K\to\infty}K^{2-\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}=0,$

this concludes the proof. Therefore, one just has to prove the Claim.

Proof of the Claim. Fix $\epsilon>0$.

$\bullet$ $G^{+}(\epsilon)$ is dense in $\mathcal{H}$. Let
$0\neq\psi\in\mathcal{H}$. Suppose now that there exists $C_{\psi}>0$ such
that for sufficiently large $K$,

$\bigg{\|}\frac{1}{K}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}\leq\frac{C_{\psi}}{K^{\epsilon/2}}$
(7)

(otherwise, $\psi\in G^{+}(\epsilon)$). It follows from Lemma 3.2 that there
exists $\eta\in\mathcal{H}$, with $\|\eta\|\leq 1$, such that

$\liminf_{K\to\infty}K^{\epsilon/3}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}=\infty.$
(8)

Finally, for each $m\in\mathbb{N}$, set
$\psi_{m}:=\psi+\displaystyle\frac{\eta}{m}$; naturally, $\psi_{m}\to\psi$ as
$m\to\infty$. Moreover, by (7) and (8), one has for each $m\in\mathbb{N}$ and
each sufficiently large $K$,

$\displaystyle
K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi_{m}-\psi_{m}^{*}\bigg{\|}^{2}$
$\displaystyle=$ $\displaystyle
K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}^{2}$
$\displaystyle+$ $\displaystyle\frac{2K^{\epsilon}}{m}{\rm
Re}\,\biggr{\langle}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*},{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\biggr{\rangle}$
$\displaystyle+$
$\displaystyle\frac{K^{\epsilon}}{m^{2}}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}$
$\displaystyle\geq$ $\displaystyle\frac{2K^{\epsilon}}{m}{\rm
Re}\,\biggr{\langle}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*},{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\biggr{\rangle}$
$\displaystyle+$
$\displaystyle\frac{K^{\epsilon}}{m^{2}}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}$
$\displaystyle\geq$
$\displaystyle-\frac{2K^{\epsilon}}{m}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi-\psi^{*}\bigg{\|}$
$\displaystyle+$
$\displaystyle\frac{K^{\epsilon}}{m^{2}}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\eta-\eta^{*}\bigg{\|}^{2}$
$\displaystyle\geq$
$\displaystyle-\frac{2K^{\epsilon}}{m}\cdot\frac{C_{\psi}}{K^{\epsilon/2}}+\frac{K^{\epsilon}}{m^{2}}\cdot\frac{1}{K^{\epsilon/3}}.$

Thus, for every $m\in\mathbb{N}$,

$\limsup_{K\to\infty}K^{\epsilon}\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi_{m}-\psi_{m}^{*}\bigg{\|}^{2}=\infty,$

from which follows that $G^{+}(\epsilon)$ is dense in $\mathcal{H}$. ∎

$\bullet$ $G^{-}(\epsilon)$ is dense in $\mathcal{H}$. Let
$\psi\in\mathcal{H}$ and set, for each $n\in\mathbb{N}$,

$A_{n}:=\\{1\\}\,\cup\,\biggr{\\{}e^{2\pi
i\theta}\,:\,\theta\in\biggr{[}\frac{1}{2n},1-\frac{1}{2n}\biggr{)}\biggr{\\}}$

and

$\psi_{n}:=P^{U}(A_{n})(\psi-\psi^{*})+\psi^{*};$

note that $\psi_{n}^{*}=\psi^{*}$ and $\psi_{n}\to\psi$ as $n\to\infty$ since
$P^{U}(A_{n})$ converges strongly to $I$ as $n\to\infty$. Moreover, by the
same arguments presented in the proof of Theorem 1.2, one obtains

$\displaystyle\liminf_{K\to\infty}K^{2-\epsilon}\,\bigg{\|}{\frac{1}{K}}\displaystyle\sum_{j=0}^{K-1}U^{j}\psi_{n}-\psi_{n}^{*}\bigg{\|}^{2}\leq\liminf_{K\to\infty}K^{2-\epsilon}\,\frac{16n^{2}\|\psi_{n}\|^{2}}{\pi^{2}K^{2}}=0,$

from which follows that $G^{-}(\epsilon)$ is dense in $\mathcal{H}$.

Acknowledgments

S. L. Carvalho thanks the partial support by FAPEMIG (Minas Gerais state
agency; Universal Project, under contract 001/17/CEX-APQ-00352-17) and C. R.
de Oliveira thanks the partial support by CNPq (a Brazilian government agency,
under contract 303689/2021-8).

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Email<EMAIL_ADDRESS>Departamento de Matemática e Estatística,
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