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ON CAYLEY IDENTITY FOR SELF-ADJOINT
OPERATORS IN HILBERT SPACES
ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
Abstract. We prove an analogue to the Cayley identity for an
arbitrary self-adjoint operator in a Hilbert space. We also provide
two new ways to characterize vectors belonging to the singular
spectral subspace in terms of the analytic properties of the resol-
vent of the operator, computed on these vectors. The latter are
analogous to those used routinely in the scattering theory for the
absolutely continuous subspace.
1. Introduction
If M is a matrix in C
n
and d
M
() := det(M) is its characteristic
polynomial, the celebrated Cayley identity says that
d
M
(M) 0.
In [25] we have studied the almost Hermitian spectral subspace of
a nonself-adjoint, non-dissipative operator L. The following criterion
has been established: a nonself-adjoint operator possesses almost Her-
mitian spectrum (i.e., its almost Hermitian spectral subspace coincides
with the Hilbert space H) i a natural generalization of Cayley identity
hold both for the operator itself and its adjoint.
This generalization of Cayley identity is formulated in terms of the
so-called weak outer annihilation. The following denition of it has
been suggested:
Denition 1.1. Let () be an outer [21] in the upper half-plane C
+
uniformly bounded scalar analytic function. We call this function a
weak annihilator of an operator L, if
w lim
0
(L + i) = 0. (1.1)
1991 Mathematics Subject Classication. Primary 47A10; Secondary 47A55.
The rst author gratefully acknowledges support received from the EPSRC grant
EP/D00022X/2. The rst and second authors acknowledge partial support from
the RFBR grants 09-01-00515-a and 11-01-90402-Ukr f a.
1
2 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
As a by-product of the aforementioned analysis of nonself-adjoint
operators and using essentially nonself-adjoint techniques (i.e., the di-
lation of a dissipative operator, see [2]) we have further been able to
prove, that a self-adjoint operator A has trivial absolutely continuous
subspace if and only if A is weakly annihilated in the sense of the above
denition.
Moreover, the corresponding outer analytic function admits an ex-
plicit choice, i.e., it can be chosen to be equal to the perturbation
determinant D
A/AiV
() of the pair A, A iV [24], where V is an
auxiliary non-negative trace class operator.
The natural question on the possibility to formulate a local version
of this criterion for self-adjoint operators with mixed spectrum, i.e.,
how to ascertain in similar terms whether the spectrum of a given self-
adjoint operator A is purely singular inside some Borel set of the real
line, was posed some time ago by Prof. David Pearson. The present
paper is an attempt to give an (in our view, so far incomplete) answer
to this question. We prove the result envisaged in two quite dierent
avours: the one that we prefer (but analytic diculties then only
allow us to give the proof under rather restrictive assumptions on the
operators spectrum, see below) and the one that actually allows to
give a rigorous proof in the most general case.
The paper is organized as follows.
Since the functional model of a nonself-adjoint operator is of crucial
importance for our approach and the proof of our main result relies
heavily upon the symmetric form of the Nagy-Foias functional model
due to Pavlov [26, 3] (see also the paper [6] by Naboko), we continue
with a brief introduction to the main concepts and results obtained in
this area in Section 2.
Section 3 contains our main result, which may be viewed as a general-
ization of the Cayley identity to self-adjoint operators with an arbitrary
spectral structure.
Finally, in Section 4 we derive two new characterizations of vectors
belonging to the singular spectral subspace of a self-adjoint operator in
terms of the analytic properties of the resolvent of the operator, com-
puted on these vectors. The latter are analogous to those used routinely
in the scattering theory for the absolutely continuous subspace.
At this time, we have elected to postpone the discussion of possible
applications, since the meaningful examples we have in mind, i.e., the
examples in which the singularity of the spectrum in a given set is either
unknown or cannot be obtained by some simpler classical techniques,
do require substantial an non-trivial analysis to be fully considered.
ON CAYLEY IDENTITY. . . 3
2. The functional model of a dissipative operator
In the present section we briey recall the functional model of a
nonself-adjoint operator constructed in [2, 3] in the dissipative case and
then extended in [4, 5, 6, 7] to the case of a wide class of non-dissipative
operators. We consider a class of nonself-adjoint operators of the form
[6] L = A + iV, where A is a self-adjoint operator in H dened on
the domain D(A) and the perturbation V admits the factorization
V = J/2, where is a non-negative self-adjoint operator in H and
J is a unitary operator in an auxiliary Hilbert space E, dened as
the closed range of the operator : E R(). This factorization
corresponds to the polar decomposition of the operator V . It can also
be easily generalized to the node case [8], where J acts in an auxiliary
Hilbert space H and V =

J/2, being an operator acting from H

to H. In order that the expression A + iV be meaningful, we impose
the condition that V be (A)-bounded with relative bound less than
1, i. e., D(A) D(V ) and for some a and b (a < 1) the condition
|V u| a|Au| + b|u|, u D(A) is satised, see [9]. Then the
operator L is well-dened on the domain D(L) = D(A).
Alongside with the operator L we are going to consider the maximal
dissipative operator L

= A + i

2
2
and the one adjoint to it, L

= Ai

2
2
. Since the functional model for the dissipative operator
L

will be used below, we require that L

is completely nonself-adjoint,
i. e., that it has no reducing self-adjoint parts. This requirement is not
restrictive in our case due to Proposition 1 in [6].
We also note that the functional model in the general case of opera-
tors with not necessarily additive imaginary part and with non-empty
resolvent set has been developed in [7].
Now we are going to briey describe a construction of the self-adjoint
dilation of the completely nonself-adjoint dissipative operator L

, fol-
lowing [2, 3], see also [6].
The characteristic function S() of the operator L

is a contractive,
analytic operator-valued function acting in the Hilbert space E, dened
for Im > 0 by
S() = I + i(L

)
1
. (2.1)
In the case of an unbounded the characteristic function is rst dened
by the latter expression on the manifold ED() and then extended by
continuity to the whole space E. The denition given above makes it
possible to consider S() for Im < 0 with S() = (S

())
1
provided
that the inverse exists at the point . Finally, S() possesses boundary
4 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
values on the real axis in the strong topology sense: S(k) S(k +
i0), k R (see [2]).
Consider the model space 1 = L
2
(
I S

S I
), which is dened in [3] (see
also [10] for description of general coordinate-free models) as Hilbert
space of two-component vector-functions ( g, g) on the axis ( g(k), g(k)
E, k R) with metric
__
g
g
_
,
_
g
g
__
=
_

__
I S

(k)
S(k) I
__
g(k)
g(k)
_
,
_
g(k)
g(k)
__
EE
dk.
It is assumed here that the set of two-component functions has been
factored by the set of elements with norm equal to zero. Although we
consider ( g, g) as a symbol only, the formal expressions g

:= ( g +S

g)
and g
+
:= (S g + g) (the motivation for the choice of notation is self-
evident from what follows) can be shown to represent some true L
2
(E)-
functions on the real line. In what follows we plan to deal mostly with
these functions.
Dene the following orthogonal subspaces in 1 :
D

_
0
H
2

(E)
_
, D
+

_
H
2
+
(E)
0
_
, K 1(D

D
+
),
where H
2
+()
(E) denotes the Hardy class [2] of analytic functions f in
the upper (lower) half-plane taking values in the Hilbert space E. These
subspaces are incoming and outgoing subspaces, respectively, in
the sense of [11].
The subspace K can be described as K = ( g, g) 1 : g

g + S

g H
2

(E), g
+
S g + g H
2
+
(E). Let P
K
be the orthogonal
projection of the space 1 onto K, then
P
K
_
g
g
_
=
_
g P
+
( g + S

g)
g P

(S g + g)
_
,
where P

are the orthogonal Riesz projections of the space L

2
(E) onto
H
2

(E).
The following Theorem holds [2, 3]:
Theorem 2.1. The operator (L

0
)
1
is unitarily equivalent to the
operator P
K
(k
0
)
1
[
K
in the space K for all
0
, Im
0
< 0.
This means, that the operator of multiplication by k in 1 serves as
a minimal (clos
Im=0
(k )
1
K = 1) self-adjoint dilation [2] of the
operator L

.
ON CAYLEY IDENTITY. . . 5
3. Characterization of singular spectrum in terms of weak
annihilation
In the present section, we attempt to provide a localized criterion
of pure singularity of the spectrum of a general self-adjoint operator
inside a given set of the real line, building upon the technique and
approach developed in [25]. It is worth mentioning that not only the
proofs of our results in this direction exploit essentially nonself-adjoint
(in particular, functional model related) techniques, but even certain
crucial objects of the nonself-adjoint spectral theory appear already in
their statements.
Our next Theorem in our view constitutes the most natural localiza-
tion of the corresponding global result of [25]. Unfortunately, we are
only able to prove the result in this natural form in the case when both
ends of the interval where one wants to ascertain pure singularity
of the spectrum are located inside a spectral gap. Any attempt to get
rid of this rather horrible restriction requiring crucial a-priori informa-
tion on the spectral structure fails due to the lack of control over the
annihilating function at the endpoints of the interval . It seems that
in the general setting one has to resort to a quite dierent (and less
natural) denition of annihilation (see Theorem 3.5 below).
Theorem 3.1. Let A be a (possibly, unbounded) self-adjoint opera-
tor in the Hilbert space H. Let a point
0
R belong together with
some neighborhood to the resolvent set of A. Then the following two
statements are equivalent.
(i) The spectrum of A to the left of the point
0
is purely singular;
(ii) There exists an outer bounded in the upper half-plane non-trivial
(i.e., non-constant) scalar function () with real boundary val-
ues almost everywhere on (
0
, +) and non-real boundary val-
ues almost everywhere on (,
0
), weakly annihilating the op-
erator A, i.e.,
w lim
0
((A+ i)

(A i)) = 0,
where

() := (

) is an outer bounded in the lower half-plane

analytic function.
Proof. Choose V to be a trace class non-negative self-adjoint operator
in the Hilbert space H such that

Im =0
(A)
1
V H = H. (3.1)
Clearly, such choice is always possible.
6 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
We follow the approach developed in [6] for the operators L admitting
the representation L

= A + /2, where 0 is a non-negative

operator in the Hilbert space H and is a bounded operator in the
subspace E, being the closure of the range of . Choose to be a
Hilbert-Schmidt class operator dened by the formula =

2V S
2
.
Then the operator L

is well-dened on the domain D(L

) = D(A).
Moreover, L

A when = 0, i.e., L
0
A. Consider the dissipative
operator L

A+iV (this operator coincides with L

iI
). Clearly, it is a
maximal dissipative operator in H; moreover, it is easy to see that the
condition (3.1) guarantees that it is also completely nonself-adjoint.
Construct the functional model based on the operator L

(see Section
1 above). In the corresponding dilation space 1 the following formulae
describe the action of the resolvent (A)
1
on all vectors ( g, g) K,
as above K being the model image of H (see [6]):
(A)
1
_
g
g
_
= P
K
1
(k )
_
g
g
1
2
_
I + (S

() I)
1
2
_
1
g

()
_
, Im < 0
(3.2)
(A)
1
_
g
g
_
= P
K
1
(k )
_
g
1
2
_
I + (S() I)
1
2
_
1
g
+
()
g
_
, Im > 0.
(3.3)
Here S() is the characteristic function of completely nonself-adjoint
maximal dissipative operator L

, all the other notation has already

been introduced above.
We introduce the following notation for the operator functions, ap-
pearing in this representation:
A
() := I +(S() I)
1
2
and

A
() :=
I + (S

() I)
1
2
. The functions
A
and

A
are bounded analytic
operator functions in half-planes C
+
and C

, respectively.
Recall that the characteristic function S() is a contraction in the
upper half plane. It follows that, since
A
() = (I + S())/2 and

A
() = (I + S

())/2, they are outer contractions (see [2]) in the

half-planes C
+
and C

, respectively.
By denition of S(), both operator functions also have well-dened
outer [19, 2] determinants
1

A
,

A
, bounded (in fact, contractive) in
their respective half-planes. It is also clear that

A
() =
A
(). We re-
mark, that
A
() is a clearly non-zero function since lim
+

A
(i) =
1.
1
It is easy to see that
A
() in fact coincides with the perturbation determinant
D
A/AiV
() of the pair A, A iV [24]
ON CAYLEY IDENTITY. . . 7
W.l.o.g. assume, that the point = 0 together with its neighborhood
belongs to the resolvent set of the operator A. It follows that since
the operator L

is completely nonself-adjoint and dissipative, the same

neighborhood also belongs to its resolvent set. Thus the function S()
admits analytic continuation to C

through the named neighborhood

of zero and the determinant
A
() is C

there.
Since () is an outer function, it admits the following representation
in terms of the logarithm of its boundary values on the real line:

A
() = e
ic
exp
_
i

_
R
_
1
t
+
t
1 + t
2
log [(t)[dt
__
,
where (t) := (t +i0) are the boundary values of the function from
above and c is some real constant. From (3.2) it further follows, that

A
(t) is separated from zero on .
Fix a point
0
such that
0
and let

1
() = e
ic
exp
_
i

_

0

_
1
t
+
t
1 + t
2
_
log [
A
(t)[dt
_
be the new outer (by construction), bounded in the upper half-plane
function. Let further (t) be the harmonic conjugate (or, in other
words, the Hilbert transform) of the function f(t), equal to log [
A
(t)[
on (,
0
) and to 0 elsewhere. Clearly, the function (t) is itself
innitely smooth on any interval [
1
, +) provided that
1
<
0
.
Choose yet another function
2
() as follows:

2
() = e
ic
exp
_
1

_
+

_
1
t
+
t
1 + t
2
_

1
(t)dt
_
,
where
1
(t) is any C
1
(R) function such that
1
(t) (t), t 0. As
it is easily seen, on the right half-line arg
2
(t + i0) = arg
1
(t + i0)
almost everywhere. Whats more, since
1
(t) is C
1
on the real line, its
harmonic conjugate is continuous and thus bounded [22]. It follows,
that the function
2
() is itself bounded and outer in the upper half-
plane, admitting the following representation:

2
() = e
ic
exp
_
i

_
+

_
1
t
+
t
1 + t
2
_
(
1
(t))dt
_
,
where
1
(t) is the harmonic conjugate of the function
1
.
Consider the function () :=
1
()
2
(). By virtue of its construc-
tion, it is a bounded outer function in the upper half-plane with almost
everywhere real boundary values on the right half-line. Whats more,
together with its rst factor it cancels out the zeroes of the function
8 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO

A
() on the interval (,
0
): [(t +i)/
A
(t +i)[ C uniformly
in for some nite constant C and every t
0
.
It remains to be seen that this function can be chosen in a way
such that its boundary values to the left of the point 0 are non-real
almost everywhere. In fact, this can be safely assumed w.l.o.g.: if
not, denote by (, 0) the set of points where the corresponding
boundary values are real almost everywhere. Then consider a non-
negative smooth enough function g(k) having its support equal to the
closure of and dene g() :=
_
g(k)
k
dk. Multiplication by this outer
bounded factor clearly equips the function () with the properties
required by the Theorem.
We will now prove that () weakly annihilates the self-adjoint op-
erator A in the sense of the Theorem.
First, let u belong to the spectral subspace E
A
(0, +)H of the oper-
ator A, where E
A
() is the operator-valued spectral measure associated
with A. Then by the spectral theorem and by Lebesgue dominated
convergence theorem it is easy to see that
lim
0
((A+i)

(Ai))u, v =
_
+
0
((k+i0) (k+i0)d
u,v
(k) = 0
for all v in H (in a nutshell, we have used the fact that the function

by its construction is an analytic continuation of the function to the

lower half-plane).
It remains to be seen that if u = E
A
(, 0)H, then
lim
0
(A+ i)u, v = 0
and
lim
0

(Ai)u, v = 0
for all v in H, provided that the spectrum of the operator A is purely
singular to the left of the point zero. We will check the rst identity
above, the second being veried analogously.
The bounded (due to v. Neumann inequality [2] or, alternatively,
due to the spectral theorem) operator (A+i) is dened by the Riesz-
Dunford integral,
(A+i)u, v =
1
2i
_
_

0
+3i/2
+3i/2

_

0
+i/2
+i/2
_

A
()

(A+ i )
1
u, v
_
d.
Using the model representation (3.2) we then immediately obtain:
_
(A+ i)
_
g
g
_
,
_

f
f
__
=
_
(k + i)
_
g
g
_
,
_

f
f
__
+
ON CAYLEY IDENTITY. . . 9
1
2i
_

0

(t+i

2
)
_
1
k (t i

2
)
_
0
1
2

1
A
(t i

2
)g

(t i

2
)
_
,
_

f
f
__
dt
1
2i
_

0

(t+i
3
2
)
_
1
k (t + i

2
)
_
1
2

1
A
(t + i

2
)g
+
(t + i

2
)
0
_
,
_

f
f
__
dt.
(3.4)
Rewriting

A
() =

()/
A
(

) and
A
() = ()/
A
() with
bounded in the lower (resp., upper) half-plane operator function

(resp., ), it is now easy to see that the last expression assumes the
following form:
_
(A+ i)
_
g
g
_
,
_

f
f
__
=
_
(k + i)
_
g
g
_
,
_

f
f
__
+
_

0

(t + i

2
)

A
(t + i

2
)
_
1
2

(t i

2
)g

(t i

2
), f
+
(t + i

2
)
_
dt
_
+

(t + i
3
2
)

A
(t + i

2
)
_
1
2
(t + i

2
)g
+
(t + i

2
), f

(t i

2
)
_
dt. (3.5)
Due to analytic properties of the functions g

2
(E), f

2
(E)
the latter expression has a limit as tends to 0 and by Lebesque dom-
inated convergence theorem and Schwartz inequality
lim
0
_
(A+ i)
_
g
g
_
,
_

f
f
__
=
_

0

[

f, g

+f, g
+
+
(t)

A
(t)

1
2
g
+
, f

+
(t)

A
(t)

1
2

, f
+
]dt.
(3.6)
Here
_
[

f, g

+f, g
+
]dt = ( g, g), (

f, f) and therefore represents
a meaningful object.
In order to prove that this limit is actually equal to zero, we recall
[19, 20] that for all ( g, g) H
s
(A) and for all (

f, f) K
_
[(L k i)
1
(L k + i)
1
]
_
g
g
_
,
_

f
f
__

0
0 (3.7)
for a. a. real k. Again taking into account formulae describing the
action of the resolvent of the operator L in the model representation
in upper and lower half-planes, consider the following expression for
arbitrary vectors ( g, g) H
s
(A), (

f, f) K H:
1
2i
(t + i)
_
[(At i)
1
(At + i)
1
]
_
g
g
_
,
_

f
f
__
=
10 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
(t + i)
2i
_

0

2i
(k t)
2
+
2
__
g
g
_
,
_

f
f
__
dk+
(t + i)

A
(t + i)
_
1
2
(t + i)g
+
(t + i), f

(t i)
_
+
(t + i)

A
(t + i)
_
1
2

(t i)g

(t i), f
+
(t + i)
_
(cf. (3.5)). The latter expression has a limit for a. a. t R, equal
to the integrand in (3.6). On the other hand, from (3.7) it follows,
that this limit is identically equal to zero for a. a. t. This observation
completes the proof.
Conversely, let the self-adjoint operator A possess a weak outer
bounded annihilator () in the sense of the Theorem. Let the vector
u ,= 0, u E(, 0)H belong to the absolutely continuous spectral
subspace H
ac
. Then, again by the spectral theorem and by Lebesgue
dominated convergence theorem it is easy to see that
_

((k + i0) (k + i0))d

u,v
(k) = 0
(by taking E
A
()v instead of v) for an arbitrary Borel set (, 0)
and the nite absolutely continuous complex measure [14] d
u,v
(k) :=
dE
A
(k)u, v, where as above E
A
is the operator valued spectral mea-
sure of the operator A and v is an arbitrary element of H. Since bound-
ary values of are non-zero almost everywhere on the real line and by
assumption these boundary values are non-real almost everywhere, this
implies that the measure d
u,v
0 for all v H.
This completes the proof.
Remark 3.2. The last Theorem can of course be easily generalized to-
gether with the proof given to the situation when the set, where one
tests the singularity of the spectrum of the operator A, is an arbitrary
nite or innite interval of the real line or even a nite unit of such
disjoint intervals.
Remark 3.3. Note that the existence of a non-zero analytic bounded
annihilator of the operator A is clearly sucient for the pure singularity
of its spectrum to the left of the point
0
. Nevertheless, our Theorem
asserts that this function can be chosen to be outer in C
+
as well.
Remark 3.4. Suppose that the operator A is a self-adjoint operator
with simple spectrum. Then the trace class operator V of the last
Theorem due to (3.1) can clearly be chosen [14] as a rank one operator
in Hilbert space H. In this situation, the proof of Theorem 3.1 can
ON CAYLEY IDENTITY. . . 11
be modied in the part concerning the choice of the annihilator in the
following way: the function
A
can be chosen as

A
() :=
1
1 i(D() 1)
,
where D() := 1 +(A)
1
, is the perturbation determinant of
the pair A, A+, and is the generating vector for the operator
A.
The proof is a straightforward application of the explicit formula
for the resolvent of a rank one perturbation of a self-adjoint operator,
based on the Hilbert identity.
We now pass over to the general case, i.e., the case when one has no
a-priori information on the spectral structure of the operator A near
the endpoints of the interval under consideration. In this case one faces
the necessity to modify somewhat the denition of annihilation. The
following Theorem addresses this.
Theorem 3.5. Let A be a (possibly, unbounded) self-adjoint operator
in the Hilbert space H. Let be an arbitrary Borel set on the real line.
Then the following two statements are equivalent.
(i) The spectrum of A in is purely singular, i.e., the intersection
of absolutely continuous spectrum and the set is empty;
(ii) There exist an outer bounded in the upper half-plane non-trivial
(i.e., non-zero) scalar function () and an outer bounded in
the upper half-plane non-constant scalar function () such that
() has non-tangential limits on the real line at every point
of the latter and these limits are zero everywhere on R and
non-zero everywhere on , weakly annihilating the operator A
in the following sense:
w lim
0
(A + i)((A+ i)

(Ai)) = 0,
where

() :=

(

) is an outer bounded in the lower half-plane

analytic function.
Proof. We start with the proof of the implication (i)(ii).
To begin with, let () be a Riesz transform of a square summable
non-negative function b(k) such that supp b = :
() =
_
b(k)
k
dk.
In order to satisfy the restrictions of the Theorem on the imaginary
part of , further assume that is in addition a C
1
function on the
real line. Then clearly it is outer bounded in the upper half-plane (in
12 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
fact, even an R-function), the imaginary part of it has boundary limits
everywhere on R [22] and moreover, these boundary limits are equal to
zero on R and are non-zero everywhere on .
Then

() is an outer bounded analytic continuation of to the

lower half-plane C

through the complement R, whereas the jump of

the continued function through , which is proportional to (k +i0),
is non-trivial everywhere on .
By the spectral theorem of a self-adjoint operator and then by the
Lebesgue dominated convergence theorem it is now easy to see that
(A + i)

(A i)
0
(A) strongly as 0, where
0
(k) :=
2i(k + i0).
On the other hand, repeating the argument from the proof of the last
Theorem (namely, from (3.4) to (3.6), where the integral is extended
from (,
0
) to the whole real line) one arrives at the conclusion
that () :=
A
(), where
A
is the same function as above, is such
that the operator family (A+i) has a weak limit as 0, given by
(3.6) with the above-mentioned change of the limits of integration.
It follows that w lim
0
(A + i)((A + i)

(A i)) exists
and its only left to prove that it is equal to zero. Let rst u E
A
().
Then (A+i)((A+i)

(Ai))u, v 0 for all v H by the

same argument as in the proof of the preceding Theorem (see (3.7) and
below).
If on the other hand u E
A
(R )H, then the named limit is zero
since
0
(A)[
E
A
(R\)H
= 0 due to the fact that (k + i0) = 0 for all
k R .
The proof of the inverse implication (ii)(i) is nothing but a slight
modication of the corresponding implication of Theorem 3.1
Indeed, let the vector u ,= 0, u E
A
()H belong to the absolutely
continuous spectral subspace H
ac
. Then, again by the spectral theorem
and by Lebesgue dominated convergence theorem it is easy to see that
_

(k + i0)((k + i0)

(k + i0))d
u,v
(k) = 0
(by taking E
A
()v instead of v) for an arbitrary Borel set . Since
boundary values of are non-zero almost everywhere on the real line
and by assumption the boundary values of the imaginary part of
are non-zero in , this implies that the absolutely continuous measure
d
u,v
0 for all v H.
This completes the proof.

ON CAYLEY IDENTITY. . . 13
4. On the analytic properties of the resolvent
We take this opportunity to prove yet another result. We begin with
the following observation, well-known from the mathematical scattering
theory. Consider a self-adjoint operator A. Then there exists a linear
set

H
a.c.
dense in the absolutely continuous spectral subspace of A such
that
_
| exp(iAt)u|
2
dt <
for all u

H
a.c.
and any non-negative operator S
2
(see, e.g., [1]).
Using the Fourier transform and Parsevals identity, its easy to see [6]
that the last condition is equivalent to:
(A)
1
u H
2

(Ran )
for all u

H
a.c.
Taking an operator V S
1
as in the proof of the
previous Theorem, i.e., a non-negative trace class operator such that
the condition (3.1) is satised, we can further obtain [6] the follow-
ing description of the absolutely continuous spectral subspace of the
operator A:
H
a.c.
= closu[

V (A)
1
u H
2

(E),
where as in Section 2 E is the auxiliary Hilbert space, being the closed
image of the operator V .
In this Section, we derive an analogous characterization for the sin-
gular spectral subspace H
s
of a self-adjoint operator A. Namely, the
following Theorem holds.
Theorem 4.1. Let A be a self-adjoint operator in the Hilbert space H.
Let V S
1
be a positive trace class operator in H such that (3.1) holds.
Then if the vector u belongs to the singular spectral subspace H
s
of A,
then the vector

V (A)
1
u belongs to vector Smirnov classes N
2

(E)
[10], i. e., it can be represented as h

()/

(), where h

H
2

(E)
and

() are scalar bounded outer analytic functions in half-planes

C

, respectively. Here the functions

can be chosen independently of

vector u.
Proof. We again use the functional model constructed based on the
dissipative operator A+ iV .
Let now u H
s
. The following identities hold (see [6]):

2g
+
() =
A
()(A)
1
u, Im > 0,

2g

() =

A
()(A)
1
u, Im < 0
(4.1)
14 ALEXANDER V. KISELEV AND SERGUEI N. NABOKO
Here the operator-functions
A
() and

A
() are dened by the iden-
tities
A
() = (I + S())/2 and

A
() = (I + S

())/2 (S being the

characteristic function of the dissipative operator A+iV ), and are outer
S
1
valued contractions in the half-planes C
+
and C

, respectively.
Within the conditions of Theorem 4.1 both operator-functions
A
()
and

A
() also possess outer determinants in their respective half-
planes [2]. Therefore, by the uniqueness theorem for scalar bounded
analytic functions [21], they are invertible for almost all real k.
Then we obtain immediately, that for all C
+

2V (A)
1
u =

2
1
A
()g
+
() =

2
1
+
()()g
+
(),
where ()
A
() =
A
()() =
+
()I with a bounded operator-
function (), i.e.,
+
is the determinant of the operator function
A
.
It remains to point out (see Section 2), that the function g
+
() belongs
to H
2
+
(E) as u H. Application of a similar argument to the vector
g

completes the proof

Remark 4.2. As it is easily seen from the denition of
A
() and

A
(),
the functions

() in the statement of Theorem 4.1 can be chosen so

that

() =

+
(

).
In a similar way we are able to give a weak version of the previous
Theorem. Indeed, one can easily ascertain (see, e.g., [1, 6]) on the
basis of F. and M. Riesz theorem [21] that the absolutely continuous
subspace of a self-adjoint operator A can be alternatively characterized
as follows:
H
a.c.
= closu[(A)
1
u, v H
2

for all v H.
The following Theorem gives an analogous representation for the sin-
gular spectral subspace.
Theorem 4.3. Let A be a self-adjoint operator in the Hilbert space H.
Then if the vector u H belongs to the singular spectral subspace H
s
,
then the function (A)
1
u, v belongs to Smirnov classes N
1

for all
v H, i. e., it can be represented as h

()/

(), where h

H
1

and

() are bounded scalar outer analytic functions in half-planes C

,
respectively. Here the functions

are independent of v H and can

be chosen independently of vector u.
Proof. We will again use the model description of the resolvent of the
operator A (3.2), from where it follows that
(A)
1
u, v =
_
1
k
_
g
g
_
,
_

f
f
_
_

_
1
k
_
1
2

1
A
()g
+
()
0
_
,
_

f
f
_
_
,
ON CAYLEY IDENTITY. . . 15
where (

f, f) is the model image of the vector v. Let u H
s
. The
rst term on the right hand side is clearly the Cauchy transformation
of an L
1
-function, whereas the second one can be rewritten by residue
calculation in the following way:
_
1
k
_
1
2

1
A
()g
+
()
0
_
,
_

f
f
_
_
= 2i
1
2

1
A
()g
+
(), f

()
E
.
By Theorem 4.1 the vector u is such that

V (A )
1
u N
2
+
(E),
and therefore by (4.1) again,
1
1
()g
+
() = h
+
()/
+
() for some
h
+
H
2
+
(E) and some outer bounded in the upper half-plane function

+
. It follows that if one puts () := 1/( + i),
(A)
1
u, v = k
1
() 2i
1

+
()

1
2
h
+
(), f

()
E

1

+
()()
[k
1
()
+
()() 2i
1
2
h
+
(), f

()
E
()] N
1
+
,
since k
1
() :=
1
k
_
g
g
_
,
_

f
f
_
and f

() H
2
+
(E).
An analogous argument applied in the case of C

completes the
proof.
Remark 4.4. Note that the functions

() appearing in the proof of

the last Theorem are the same as in the proof of Theorem 4.1, i.e., these
can be chosen to be equal to the determinants of the operator-functions

A
() and

A
(), respectively. Thus, the corresponding outer factors
in Theorems 4.1 and 4.3 admit the simplest (and explicitly computable)
form in the situation when the spectrum of the operator A is simple,
see Remark 3.4.
Acknowledgements. Both authors express their gratitude to Prof.
David Pearson for the interest expressed by him to their research and
for the question that motivated this paper.
The rst author is grateful to Prof. A. Sobolev for fruitful discussions
during the authors stay in UCL.
The rst author is grateful to the Dept. of Mathematics, University
College London where parts of this work were done for hospitality.
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Department of Higher Mathematics and Mathematical Physics, St.Petersburg
State University, 1 Ulianovskaya Street, St.Petersburg, St. Peter-
hoff 198504 Russia
Department of Higher Mathematics and Mathematical Physics, St.Petersburg
State University, 1 Ulianovskaya Street, St.Petersburg, St. Peter-
hoff 198504 Russia
E-mail address: alexander.v.kiselev@gmail.com