Difference between revisions of "Nuclear norm"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> | ||
− | where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784023.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784024.png" /> with this norm is a Banach space that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784025.png" /> as a dense linear subspace. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784026.png" />, then the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784027.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784030.png" /> denote the usual operator norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784037.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784041.png" />. Any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784042.png" /> can be represented in the form | + | where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784023.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784024.png" /> with this norm is a Banach space that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784025.png" /> as a dense linear subspace. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784026.png" />, then the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784027.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784030.png" /> denote the usual [[operator norm]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784037.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784041.png" />. Any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784042.png" /> can be represented in the form |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table> |
Revision as of 17:13, 29 October 2017
trace norm
A norm on the space of nuclear operators (cf. Nuclear operator) mapping a Banach space
into a Banach space
.
Let and
be Banach spaces over the field of real or complex numbers, let
be the space of all continuous linear operators mapping
into
, and let
be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of
is denoted by
, and the value of a functional
at a vector
by
.
Every nuclear operator can be represented in the form
![]() | (1) |
where and
are sequences in
and
, respectively, such that
![]() |
such representations are called nuclear. The quantity
![]() | (2) |
where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of . The space
with this norm is a Banach space that contains
as a dense linear subspace. If
, then the adjoint operator
belongs to
, and
. Let
denote the usual operator norm in
. Then
for all
. If
and
, then
, and
; if
and
, then
, and
. Any operator
can be represented in the form
![]() | (3) |
The quantity
![]() | (4) |
where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of . The space
can be identified with the tensor product
. Here, to an operator
of the form (3) there corresponds the element
![]() | (5) |
and the finite nuclear norm (4) goes into the norm
![]() | (6) |
where the infimum is taken over all finite representations of in the form (5). This norm is called the tensor (or cross) product of the norms in
and in
. The completion of
with respect to the norm (6) is denoted by
. The mapping
, under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator
. The range of
is
. If
establishes a one-to-one correspondence between
and
, then
coincides with the closure of
with respect to the norm (4); in this case the restriction of the nuclear norm to
is the same as the finite nuclear norm. But, in general,
may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in
(see Nuclear operator).
Let , where
is a separable Hilbert space, let
be the algebra of bounded operators on
, and let
be the ideal of nuclear operators in
. In this case
is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each
has a trace
(see Nuclear operator). The nuclear norm of an operator
coincides with
, where
is the adjoint of
in
. The nuclear norm is connected with the Hilbert–Schmidt norm
by
. The general form of a continuous linear functional on the Banach space
is given by
![]() | (7) |
where is an arbitrary operator from
, and the norm of the functional (7) coincides with
. Consequently,
is isometric to the dual of
. Formula (7) also gives the general form of a linear functional on the closed subspace
of
that consists of all completely-continuous (compact) operators; here
and
ranges over
. In this case the norm of the functional (7) coincides with
, that is, the space
of nuclear operators with the nuclear norm is isometric to the dual of
in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.
Example. Let be the space of summable sequences. An operator
is contained in
if and only if there is an infinite matrix
such that
sends
to
, and
. In this case,
.
References
[1] | A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955) |
[2] | A. Pietsch, "Operator ideals" , North-Holland (1980) |
[3] | A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German) |
[4] | I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian) |
[5] | I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian) |
[6] | K. Maurin, "Methods of Hilbert spaces" , PWN (1967) |
[7] | M.M. Day, "Normed linear spaces" , Springer (1958) |
Comments
References
[a1] | A. Pietsch, "Eigenvalues and ![]() |
[a2] | A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques" Bol. Soc. Mat. São Paulo , 8 (1956) pp. 1–79 |
[a3] | H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German) |
Nuclear norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuclear_norm&oldid=42221