on hilbert schmidt and schatten p class operators in p-adic hilbert
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On Hilbert Schmidt and Schatten P Class
Operators in P-adic Hilbert Spaces
SUDESHNA BASU
Department of Mathematics
George Washington University
USA
1
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In this talk, we introduce the well known
Hilbert Schimdt and Schatten-P class
operators on p-adic Hilbert spaces. We also
show that the Trace class operators in
p-adic Hilbert spaces contains the class of
completely continuous operators, which con-
tains the Schatten Class operators.
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Let K be a complete ultrametric valued field. Clas-
sical examples of such a field include the field Qp of
p-adic numbers where p is a prime, and its various
extensions .
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An ultrametric Banach space E over K is said
to be a free Banach space if there exists a fam-
ily (ei)i∈I of elements of E such that each element
x ∈ E can be written uniquely as x =∑
i∈I xiei
that is, limi∈I xiei = 0 and ‖x‖ = supi∈I |xi| ‖ei‖
Such (ei)i∈I is called an “orthogonal base” for
E, and if ‖ei‖ = 1, for all i ∈ I , then (ei)i∈I is
called an “orthonormal base”.
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Throughout this discussion, we consider free Ba-
nach spaces over K and we shall assume that the
index set I is the set of natural numbers N.
For a free Banach space E, let E∗denote its topo-
logical dual and B(E) the set of all bounded linear
operators on E. Both E∗ and B(E) are endowed
with their respective usual norms.
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For (u, v) ∈ E × E∗, we define (v ⊗ u) by:
∀x ∈ E, (v ⊗ u) (x) = v (x) u = 〈v, x〉u,
then (v ⊗ u) ∈ B(E) and ‖v ⊗ u‖ = ‖v‖ . ‖u‖.
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Let (ei)i∈N be an orthogonal base for E, then one
can define e′i ∈ E∗ by:
x =∑
i∈N
xiei, e′i (x) = xi.
It turns out that ‖e′i‖ =1
‖ei‖, furthermore, every
x′ ∈ E∗ can be expressed as x
′=∑i∈N
〈x′, ei〉 e′i and
‖x′‖ = supi∈N
|〈x′, ei〉|‖ei‖
.
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Each operator A on E can be expressed as a point-
wise convergent series, that is, there exists an infinite
matrix (aij)(i,j)∈NxNwith coefficients in K, such that:
A =∑
ij aij(e′j ⊗ ei),
and for any j ∈ N , limi→∞
|aij| ‖ei‖ = 0.
Moreover, for any s ∈ N
Aes =∑i∈N
aisei and ‖A‖ = supi,j
|aij| ‖ei‖‖ej‖
.
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Let ω = (ωi)i∈I be a sequence of non-zero elements
in the valued field K and
Eω ={x = (xi)i∈N | ∀i, xi ∈ K and lim
i→∞|xi| |ωi|1/2 = 0
}.
Then, it is easy to see that x = (xi)i∈N ∈ Eω if and
only if limi→∞
x2iωi = 0. The space Eω is a free Banach
space over K, with the norm given by:
x = (xi)i∈N ∈ Eω, ‖x‖ = supi∈N
|xi| |ωi|1/2 .
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In fact, Eω is a free Banach space and it admits a
canonical orthogonal base, namely, (ei)i∈N , where
ei = (δij)j∈N , where δij is the usual Kronecker sym-
bol. We note that for each i, ‖ei‖ = |ωi|1/2 . If
|ωi| = 1, we shall refer to (ei)i∈N as the canonical
orthonormal base.
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Let 〈, 〉 : Eω × Eω → K be defined by: ∀x, y ∈
Eω, x = (xi)i∈N , y = (yi)i∈N , 〈x, y〉 =∑i∈N
ωixiyi.
Then, 〈, 〉 is a symmetric, bilinear, non-degenerate
form on Eω, with value in K, The space Eω endowed
with this form 〈, 〉 is called a p-adic Hilbert space.
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It is not difficult to see that this “inner product”
satisfies the Cauchy-Schwarz-Bunyakovsky inequal-
ity:
x, y ∈ Eω, |〈x, y〉| ≤ ‖x‖ . ‖y‖ .
and on the cannonical orthogonal base, we have:
〈ei, ej〉 = ωiδij =
0, if i 6= j
ωi, if i = j.
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In sharp contrast to classical Hilbert spaces, in p-
adic Hilbert space, there exists x ∈ Eω, such that
|〈x, x〉| = 0 while ‖x‖ 6= 0.
Again in sharp contrast to classical Hilbert space,
all operators in B(Eω) may not have an adjoint.
So, we denote by
B0(Eω) = {A ∈ B(Eω) : ∃A∗ ∈ B(Eω)}.
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It is well known that the operator
A =∑
ij aij(e′j⊗ei) ∈ B0(Eω) ⇐⇒ ∀i, lim
j→∞
|aij||ωj|1/2
=
0, and A∗ =∑
ij ω−1i ωjaji(e
′j ⊗ ei)
The space B0(Eω) is stable under the operation of
taking an adjoint and for any A ∈ B0(Eω) : (A∗)∗ =
A and ‖A‖ = ‖A∗‖ .
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1. Schatten-p class Operators
Let us now recall few well known results from the
Scahtten class operators in classical Hilbert spaces.
Let H be a Hilbert space and let T : H −→ H
be a compact operator. Then the operator |T |, de-
fined as |T | = (TT ∗)12 is also a compact operator
hence its spectrum consists of at most countably
many distinct Eigenvalues. Let σj(|T |) denote the
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eigen values of |T |. These numbers are called the
singular numbers of T . Let (σj(|T |) denote the non
increasing sequence of the singular numbers of T ,
every number counted according to its multiplicity
as an eigenvalue of |T |.
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Definition 1.1. For 0 < r < ∞, let Br(H) de-
note the following
Br(H) = {T ∈ B(H) :∑
j
σ(j)(|T |r) < ∞|}
Then Br(H) is called the Scahtten-r-class opera-
tors of H.
The difficulty that arises in defining a Scahtten r-
class operator in a p-adic Hilbert space is there is
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no well developed theory of compact operator and
its representation and there is no notion of positive
operators existing in the literature, the reason be-
ing the inner product is defined on the abstract ul-
tramteic field K, as opposed to Real or Complex
fields. So, to introduce this class in p-adic Hilbert
spaces we first look at the following straightforward
observation for Br(H). For more details, see [14].
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Proposition 1.2. Let Br(H) denote the Schat-
ten r-class operators of H and let {ej} be the
eigenvectors corresponding to the eigenvalues{σj(|T |)}.
Then T ∈ Br(H) if and only if∑
j ‖T (ej)‖r < ∞.
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Definition 1.3. An operator A ∈ B(Eω) is said
to be in Schatten-p class ( 1 ≤ p < ∞) denoted
by Bp(Eω) if
‖A‖p = (∑ ‖A(es)‖p
|ωs|p2
)1p < ∞.
Remark 1.4. If p = 2, we get the p-adic Hilbert
Schimdt operator.
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Proposition 1.5. If A ∈ B2(Eω), then
(i) A has an adjoint A∗.
(ii) ‖A‖2 = ‖A∗‖2 and A∗ ∈ B2(Eω). In particu-
lar, B2(Eω) ⊆ B0(Eω).
(iii) ‖A‖ ≤ ‖A‖2
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Example 1.6. Suppose K = Qp. . Also, let, ωi =
pi+1, hence |ωi| = p−i−1 → 0 as i → ∞.
We define an operator A on Eω as, A =∑
aije′j⊗
ei where, aij = ωji
Then, clearly limi |aij|‖ei‖ = 0, hence A ∈ B(Eω)
Also, ‖A(es)‖ = ‖∑∞
i=0 aisei‖ = supi1
pis+s+i+1 =
1p2s+2 Hence ‖A(es)‖
|ωs|12
= ps+12
p2s+2 Therefore,∑∞
s=1‖A(es)‖r
|ωs|r2
=
∑1
p3r(s+1)
2
< ∞. Hence A ∈ Br(Eω).
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For every A ∈ Bp(Eω), we define its Schatten-p
norm as
|||A||| = ‖A‖p
In the space B2(Eω), we introduce a symmetric bi-
linear form, namelyA, B ∈ B2(Eω), 〈A,B〉 =
∑s
〈Aes, Bes〉ωs
. Its relationship with the Hilbert-Schmidt
norm is through the Cauchy-Schwarz-Bunyakovsky
inequality.
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Theorem 1.7. A, B ∈ B2(Eω), |〈A, B〉| ≤ |||A||| . |||B||| .
Proof: A =∑i,j
aij
(e′j ⊗ ei
)and B =
∑i,j
bij
(e′j ⊗ ei
),
then | < A, B > | =
∣∣∣∣∑s
〈Aes, Bes〉ωs
∣∣∣∣ ≤ sups
∣∣∣∣〈Aes, Bes〉
ωs
∣∣∣∣
≤ sups
‖Aes‖ ‖Bes‖|ωs|
≤(
sups
‖Aes‖|ωs|1/2
).
(sups
‖Bes‖|ωs|1/2
)
= sups
‖Aes‖‖es‖
. sups
‖Bes‖‖es‖
= ‖A‖ . ‖B‖ ≤ |||A||| . |||B||| �
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Proposition 1.8. For any padic Hilbert space
Eω, the following is true
(i) B2(Eω) is a two ideal of B0(Eω).
(ii) If S, T ∈ B2(Eω), then ST ∈ B2(Eω) and
‖|ST |‖ ≤ ‖|S|‖ ‖|T |‖, i. e. B2(Eω) is a
normed algebra with respect to the Hilbert
Schmidt norm
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2. Completely Continuous operators
and Schatten -p class
Definition 2.1. An operator A ∈ B(Eω) is com-
pletely continuous if it is the limit, in B(Eω), (i.e.a
uniform limit) of a sequence of operators of finite
ranks. We denote by C(Eω) the subspace of all
completely continuous operators on Eω.
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Proposition 2.2. Every Scahtten-p class opera-
tor is completely continuous,
i.e, Bp(Eω) ⊂ C(Eω).
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3. Trace and Schatten-p class
One important notion in the classical theory is that
of trace.
Definition 3.1. For A ∈ L (Eω) , we define the
trace of A to beTrA =∑s
〈Aes, es〉ωs
if this series
converges in K. We denote by T C (Eω) the sub-
space of all Trace class operators, namely, those
operators for which the trace exists.
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Remark 3.2. Let A =∑i,j
aij
(e′j ⊗ ei
)and sup-
pose that A ∈ T C (Eω) then, the series∑k
akk
converges and TrA =∑k
akk.
Theorem 3.3. Bp(Eω) ⊂ T C (Eω), i.e., every
Scahtten-p Class operator has a trace.
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Theorem 3.4. B2(Eω) ⊂ T (Eω), i.e., every Hilbert-
Schmidt operator has a trace.
Proof. Let A =∑i,j
aij
(e′j ⊗ ei
)be a Hilbert-Schmidt
operator.∑k
‖Aek‖2
|ωk|converges, hence, lim
k
‖Aek‖2
|ωk|=
0. We observe that for any k, |akk|2 |ωk| ≤ supi|aik|2 |ωi| =
‖Aek‖2, therefore, |akk|2 ≤ ‖Aek‖2
|ωk|, which implies
that limk|akk| = 0, i.e.lim akk = 0 and the series
k
∑k
akk
converges in K. �
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Remark 3.5. The above theorem is in sharp con-
trast with the classical case, since ∃ Schatten-p
Class operators which do not have traces. In fact,
for a classical Hilbert space H, the trace class op-
erator B1(H) is a subset of the algebra of Hilbert
Schimdt operators in H. In the padic case how-
ever, we definitely have B1(Eω) ⊆ B2(Eω) but the
class of operators with a trace is a larger class.
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We also have the following
Theorem 3.6. Suppose A ∈ Bp(Eω) and B ∈
B0(Eω), then Tr(AB) = Tr(BA).
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4. Examples
Example 4.1. Assume that the filed K = Qp
and consider the linear operator on Eω defined by
Aes =∑+∞
k=0 aksek,where aks = ps+k
1+p+p2+....+ps if k ≤
s and aks = 0 if k > s. Suppose that |ωk| ≥ 1
for all k and that supk |ωk| ≤ M for some posi-
tive real number M , then, the operator A defined
above is in Bp(Eω).
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Example 4.2. K = Qp and suppose that |ωs| →
∞ as s → ∞.
For integers m ≥ 1 and n ≥ 0, let
A(m,n) =∑
i.j
1
ωmi ωn
j
(e′j ⊗ ei
).
Then, A(m,n) ∈ Bp (Eω) .
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Example 4.3. Assume that the series∑s|ωs| con-
verges and fix a vector x = (xs)s∈N ∈ Eω. Let A
be such that Aes = 〈x, es〉 es = xsωses. Then
A ∈ B2 (Eω) . Now, A =∑i,j
(δijxjωj)(e′j ⊗ ei
).
Since lims|ωs| = 0, then A ∈ B (Eω).
Moreover,‖Aes‖2
|ωs|= |xs|2 |ωs|2
= |〈x, es〉|2
≤ ‖x‖2 |ωs| , and hence A ∈ B2 (Eω).
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