On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors
For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficie...
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| Date: | 2005 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2005
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509724797042688 |
|---|---|
| author | Rabanovych, V. I. Рабанович, В. І. |
| author_facet | Rabanovych, V. I. Рабанович, В. І. |
| author_sort | Rabanovych, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:42Z |
| description | For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients. |
| first_indexed | 2026-03-24T02:45:39Z |
| format | Article |
| fulltext |
UDK 517.98
V. I. Rabanovyç (In-t matematyky NAN Ukra]ny, Ky]v)
PRO ROZKLAD DIAHONAL|NOHO OPERATORA
V LINIJNU KOMBINACIG IDEMPOTENTIV
ABO PROEKTORIV
*
We prove that a bounded operator is a linear combination of three idempotents if it is not a sum of scalar
and compact operators and is similar to a diagonal one. We also prove that any self-adjoint diagonal
operator is a linear combination of four orthoprojections with real coefficients.
Dovedeno, wo obmeΩenyj operator, qkyj ne [ sumog skalqrnoho i kompaktnoho operatoriv i po-
dibnyj do diahonal\noho, [ linijnog kombinaci[g tr\ox idempotentiv, a bud\-qkyj samosprqΩe-
nyj diahonal\nyj operator [ linijnog kombinaci[g çotyr\ox ortoproektoriv iz dijsnymy koefi-
ci[ntamy.
Vstup. U roboti otrymano rozklad diahonal\noho obmeΩenoho operatora D v
separabel\nomu hil\bertovomu prostori H v linijnu kombinacig idempotentnyx
operatoriv Qi, Qi
2
= Qi, ta qkwo D*
= D, to v linijnu kombinacig ortoproek-
toriv (proektoriv) Pi, Pi
2
= Pi
∗
= Pi . Dosyt\ povnyj ohlqd po cij tematyci
moΩna znajty v [1]. Qk pokazano v [2], bud\-qkyj obmeΩenyj operator [ sumog
p’qty idempotentiv. Bil\ß toho, qkwo operator A ≠ λ I + K ( I — odynyçnyj, K
— kompaktnyj), to A [ sumog çotyr\ox idempotentiv. Na pidstavi rezul\tativ
robit [2, 3] lehko pokazaty, wo 2 I + K takoΩ [ sumog çotyr\ox idempotentiv.
Ta-kym çynom, koΩen operator [ linijnog kombinaci[g çotyr\ox idempotentiv
(ale ne sumog çotyr\ox idempotentiv [4]). Newodavno avtorom v [5] bulo
pokazano, wo koΩna skinçenna matrycq [ linijnog kombinaci[g tr\ox
idempotentnyx matryc\. U p.81 my pokaΩemo, wo operatory, podibni do
diahonal\nyx, [ linijnog kombinaci[g tr\ox idempotentiv, qkwo ]x vyhlqd
vidminnyj vid λ I + K. Nam nevidomo, çy isnugt\ operatory, qki ne [ linijnog
kombinaci[g tr\ox idempotentiv.
U 1984 r. K. Matsumoto doviv, wo bud\-qkyj samosprqΩenyj operator [ linij-
nog kombinaci[g p’qty ortoproektoriv [6]. Krim toho, qkwo prostir skinçenno-
vymirnyj, to dostatn\o linijno] kombinaci] çotyr\ox proektoriv [7]. My dovede-
mo (teorema82), wo diahonal\nyj samosprqΩenyj operator [ linijnog kombinaci-
[g çotyr\ox ortoproektoriv. Qk i dlq idempotentnoho vypadku, my ne zna[mo,
çy isnugt\ operatory, qki ne [ linijnog kombinaci[g çotyr\ox ortoproektoriv.
Dali budemo vykorystovuvaty poznaçennq I dlq odynyçnoho operatora i A ≈
≈ B dlq podibnosti miΩ operatoramy A i B : A = C
–
1
B C, de C i C
–
1
— obme-
Ωeni operatory. Matrycg obmeΩenoho operatora v fiksovanomu ortonormova-
nomu bazysi, u qko] lyße diahonal\ sklada[t\sq z nenul\ovyx elementiv a1, a2,
a3, … , budemo poznaçaty qk diag ( a1, a2, a3, … ) .
1.$$Rozklad diahonal\nyx operatoriv u linijnu kombinacig tr\ox idempo-
tentiv. U c\omu punkti my rozhlqda[mo diahonal\ni operatory, tobto operato-
ry vyhlqdu C
–
1 diag ( a1, a2, a3, … ) C, de C, C
–
1
— obmeΩeni operatory. Poç-
nemo z dovedennq isnuvannq rozkladu v linijnu kombinacig tr\ox idempotentiv
dlq operatoriv zi special\noho klasu.
Lema$1. Nexaj a1, … , an , … i b1, … , bn , … — dvi zbiΩni poslidovnosti
kompleksnyx çysel i isnu[ çyslo ε > 0 take, wo vykonugt\sq nerivnosti | ai –
– bi | > ε dlq vsix i ∈ N,
* Çastkovo pidtrymano hrantom Prezydenta Ukra]ny (# F8/320-2004) i Fondom fundamental\-
nyx doslidΩen\ Ukra]ny (proekt # 01.07/071).
© V. I. RABANOVYÇ, 2005
388 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRO ROZKLAD DIAHONAL|NOHO OPERATORA V LINIJNU KOMBINACIG … 389
i
i n na a
=
∞
→∞∑ −
1
lim < ∞ i
i
i n nb b
=
∞
→∞∑ −
1
lim < ∞ .
Todi operator D = diag ( a1, b1, a2, b2, a3, b3, … ) [ linijnog kombinaci[g tr\ox
idempotentnyx operatoriv.
Dovedennq. Zapyßemo konstanty, na qkyx bude bazuvatysq pobudova idem-
potentnyx operatoriv:
c = 10 1
1 1i
i n n
i
i n n
j
j ja a b b a b
=
∞
→∞ =
∞
→∞ ∈
∑ ∑− + − + +( ) +
lim lim sup
N
,
λ1 = 2 c, λ2 = lim lim
n n n na b
→∞ →∞
+ − −1 1λ , λ3 = 1.
Naßa meta — dovesty, wo D podibnyj do operatora, qkyj [ linijnog kombina-
ci[g λ1 Q1 + λ2 Q2 + λ3 Q3, de Qi — idempotenty. Dlq c\oho vvedemo poslidov-
nist\ xj :
x1 = λ1 ,
x j2 = a b xj j j+ − − −1 2 1 =
i
j
i ia b j
=
∑ + − + + +
1
1 2 3 2( ) ( )λ λ λ λ ,
x j2 1+ = λ λ1 2 2+ − x j . (1)
U cij poslidovnosti elementy x j2 [ „blyz\kymy” do çysla λ2, a elementy
x j2 1− — do çysla – λ2. Spravdi,
x j2 2− λ =
i
j
i n n i n na a b b
= →∞ →∞∑ − + −
1
lim lim ≤
c
10
(2)
i
x j2 1 2+ + λ = λ λ λ1 2 2 2+ + −( )x j = lim lim
n n n n ja b x
→∞ →∞
+ − + −1 2 2λ ≤
c
5
. (3)
Qk naslidok, vyvodymo, wo v poslidovnosti x1, x2, … susidni çysla [ riznymy i,
bil\ß toho,
x xj j+ −1 > c. (4)
Qkwo poklasty z i = ( )( ) / ( )x x a b a bi i i i i i2 1 2 1− − − − , to nastupna riznycq diaho-
nal\no] ta idempotentno] matryc\ bude maty vlasni çysla x i2 1− i x i2 :
Vi =
a
b
z z
z z
i
i
i i
i i
0
0 1 1
−
− −
≈
x
x
i
i
2 1
2
0
0
−
. (5)
Operator �
1
∞
Vi podibnyj do operatora X = diag ( x1, x2, x3, … ) , tobto
�
1
∞
Vi = C XC−1 . (6)
Pry c\omu na pidstavi nerivnostej (2) – (4) i a bi i− > ε moΩna vvaΩaty, wo
normy operatoriv C i C
–
1
obmeΩeni çyslom Ω1, qke zaleΩyt\ til\ky vid çy-
sel sup | aj | , sup | bj | , sup | xj | , c i ε . Iz (5) i (6) bezposeredn\o vyplyva[, wo
D ≈ X + Q3 , (7)
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
390 V. I. RABANOVYÇ
de Q3 — idempotent. Krim toho, isnugt\ idempotentni operatory Q1 i Q2 taki,
wo
X = λ1 Q1 + λ2 Q2 . (8)
Spravdi, pry 2x ≠ λ1 + λ2 diahonal\na matrycq
x
x
0
0 1 2λ λ+ −
≈ λ λ λ
λ λ
1
1 1
1 1
2 2
1
2
1
2
x x
x x− −
+ λ λ λ
λ λ
2
2 2
2 2
2 2
2
1 1
2
x x
x x
−
− −
[ fiksovanog linijnog kombinaci[g dvox idempotentiv. OtΩe, na pidstavi rivno-
sti (1) koΩna matrycq Ui = diag ( x2i, x2i + 1 ) [ linijnog kombinaci[g dvox idem-
potentiv:
Ui = λ1 1Q i( ) + λ2 2Q i( )
.
Iz vlastyvosti (4) vyplyva[, wo Q i
1
( ) i Q i
2
( ) , i ≥ 1, moΩna vybraty tak, wob ]x
normy buly obmeΩeni konstantog Ω2, qka zaleΩyt\ til\ky vid c i sup | xj | .
Tomu dostatn\o poklasty
Q1 =
( ) ( )1
1
1� �
∞
Q i , Q2 =
( ) ( )0
1
2� �
∞
Q i ,
i otryma[mo rivnist\ (8), qka razom iz (7) dovodyt\ lemu81.
ZauvaΩennq$1. Zbil\ßennq znaçennq konstanty c v dovedenni lemy81 ne
vplyva[ na spravedlyvist\ ]] dovedennq. TakoΩ zrozumilo, wo normy pobudova-
nyx idempotentnyx operatoriv zaleΩat\ til\ky vid konstant c i ε.
Teorema$1. Diahonal\nyj operator D ≠ λ I + K [ linijnog kombinaci[g
tr\ox idempotentnyx operatoriv. Podibnyj do n\oho operator takoΩ [ linij-
nog kombinaci[g tr\ox idempotentiv.
Dovedennq. Nexaj D = diag ( d1, … , dn , … ) i D ne [ operatorom vyhlqdu
λ I + K. Vyberemo v poslidovnosti di, i ≥ 1, dvi zbiΩni pidposlidovnosti dlk
i
dmk
tak, wob limk ld
k→∞ – limk md
k→∞ = δ ≠ 0 ta
∀ j ∈ N d dl k lj k
−
→∞
lim <
δ
4
i d dm k mj k
−
→∞
lim <
δ
4
. (9)
Bez obmeΩennq zahal\nosti moΩna vvaΩaty, wo mnoΩyna
Ñ = n k n l n mk k∈ ∀ ∈ ≠ ≠{ }N N : ,
[ neskinçennog. Poznaçymo ak : = dlk
, bk : = dmk
ta perepoznaçymo vsi ele-
menty z poslidovnosti di, i ∈ Ñ , qk c1, c2, c3, … . Zafiksu[mo odne z rozbyttiv
N na nezliçenni mnoΩyny:
N = N1 ∪ N2 ∪ … ∪ Np ∪ … ,
de elementy ki
p
z Np vporqdkovani za nyΩnim indeksom: k p
1 < k p
2 < k p
3 < … .
Todi za lemog81 operator Di vyhlqdu
diag c a b a bi k k k ki i i i, , , , ,
1 1 2 2
…( ), qkwo c ai ki−
1
>
δ
4
,
diag c b a b ai k k k ki i i i, , , , ,
1 1 2 2
…( ), qkwo c ai ki−
1
≤
δ
4
,
[ linijnog kombinaci[g tr\ox idempotentiv. Bil\ß toho, oskil\ky D obmeΩe-
nyj i vykonu[t\sq (9), to konstanty c i ε v lemi81 moΩna vybyraty [dynymy dlq
vsix operatoriv D1, D2, D3, … . OtΩe,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRO ROZKLAD DIAHONAL|NOHO OPERATORA V LINIJNU KOMBINACIG … 391
Ds = λ1 1Q s( ) + λ2 2Q s( ) + λ3 3Q s( )
,
i normy operatoriv Q s
1
( ) , Q s
2
( ) , Q s
3
( ) obmeΩeni çyslom, qke ne zaleΩyt\ vid zna-
çennq s (dyv. zauvaΩennq81). Za pobudovog
�
1
∞
Di ≈ D,
zvidky bezposeredn\o vyplyva[ isnuvannq rozkladu operatora D v linijnu kom-
binacig tr\ox idempotentiv.
Oskil\ky operator, podibnyj do idempotentnoho, [ idempotentom, to teore-
ma81 spravdΩu[t\sq i dlq operatoriv, podibnyx do diahonal\nyx.
Teoremu81 dovedeno.
2. Rozklad samosprqΩenyx diahonal\nyx operatoriv u linijnu kombina-
cig çotyr\ox proektoriv. Dali zapys A ≈u B bude oznaçaty, wo isnu[ unitar-
nyj operator U takyj, wo A = U BU−1 . Perß niΩ dovodyty osnovnu teoremu
punktu, navedemo proste texniçne tverdΩennq dlq ( 2 × 2 ) -matryc\ i dovedemo
lemu82, qka [ çastkovym vypadkom teoremy82.
TverdΩennq$1. Nexaj a, b ∈ R, a < 0 < b i x ∈ [ a, b ] . Todi pry | x | ≥
≥ 8| a + b | matrycq diag ( x, a + b – x ) = a P1 + b P2, de P1 i P2 — deqki orto-
proektorni matryci.
Dovedennq tverdΩennq proste i my joho ne navodymo.
Lema$2. Nexaj a1 ≥ a2 ≥ … ≥ an ≥ … — monotonna poslidovnist\ dijsnyx
çysel i konstanta
F =
i
i n na a
=
∞
→∞∑ −
1
lim < ∞ .
Todi dlq bud\-qkoho b ∈ R operator D = diag ( b, a1, a2, a3, … ) [ dijsnog li-
nijnog kombinaci[g çotyr\ox ortoproektoriv.
Dovedennq. Ideq dovedennq polqha[ v pobudovi diahonal\nyx operatoriv
X8 = diag ( x1, x2, x3, … ) i Y = diag ( y1, y2, y3, … ) takyx, wob
X = λ1 P1 + λ2 P2, Y = λ3 P3 + λ4 P4 i D ≈u X + Y,
de Pi — proektory. Rozib’[mo joho na try çastyny.
1. Vyznaçennq λj i xi . Poklademo
c = 10 1F b a
i
i+ + +
∈
sup
N
,
λ1 = 2 c, λ2 = 2 1lim
n na
→∞
− λ , λ3 = 4 c, λ4 = – 4 c.
Qkwo
b a F
n n− +
→∞
lim > 0, (10)
to isnu[ k ∈ N take, wo
b a a a
n n
i
k
i n n− + −
→∞ =
−
→∞∑lim lim
1
2 1
> 0. (11)
Nexaj pry nevykonanni umovy (10) k : = 1. U zaleΩnosti vid znaçennq b vvody-
mo poslidovnist\ çysel x1, x2, x3, … :
x1 = λ1 , qkwo vykonu[t\sq nerivnist\ (10), i x1 = λ2 — v inßyx vypadkax,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
392 V. I. RABANOVYÇ
x j2 1+ = λ λ1 2 2+ − x j , j ∈ N, (12)
x j2 = a a xj j j2 1 2 2 1− −+ − =
i
j
ia j x
=
∑ − − + −
1
2
1 2 11( )( )λ λ , j = 1, … , k – 1,
x k2 = a b xk k2 1 2 1− −+ − ,
x l2 = a a xl l l2 2 2 1 2 1− − −+ − = b a l x
i
l
i+ − − + −
=
−
∑
1
2 1
1 2 11( )( )λ λ , l > k .
2. Vlastyvosti xi . Za vyznaçennqm λ λ1 2+ = 2 limn na→∞ , zvidky
x j2 =
i
j
i n na a
= →∞∑ −
+
1
2
2lim λ , j < k ,
abo
x l2 = b a a a x
n n
i
l
i n n− + −
+ + −
→∞ =
−
→∞∑lim lim
1
2 1
1 2 1λ λ , l ≥ k.
Pry vykonanni nerivnostej (10) i (11) ma[mo x j2 ≥ λ2 i x l2 ≥ λ λ λ1 2 1+ − = λ2 ,
u protyvnomu razi x l2 ≤ λ λ λ1 2 2+ − = λ1 , l ≥ 1. Qk i pry dovedenni lemy81,
pokaΩemo, wo elementy x j2 [ „blyz\kymy” do çysla – x1, a elementy x j2 1+ —
do çysla x1 :
x xi2 1+ =
j
i
j n n n n
n n
j
i
j n n n n
a a a c i k
b a a a a c i k
= →∞ →∞
→∞ =
−
→∞ →∞
∑
∑
−
+ ≤ <
− + −
+ ≤ ≥
1
2
1
2 1
2 5
2 10
lim lim ,
lim lim lim ,
/
/
pry
pry
x xi2 1 1+ − = λ λ1 2 2 1+ − +( )x xi = 2 2 1lim ( )
n n ia x x
→∞
− + ≤ 2 5c/ , i ∈ N .
Qk naslidok, otrymu[mo
x xi i+ −1 > c i xi ∈ [ λ2, λ1 ] .
3. Pobudova proektoriv Pj . Operator X = diag ( x1, x2, x3, … ) [ linijnog
kombinaci[g dvox proektoriv P1 ta P2 :
X = λ1 P1 + λ2 P2 . (13)
Wob dovesty vlastyvist\ (13), rozklademo matrycg X u prqmu sumu matryc\:
X = ( ) ( , ) ( , )x x x x xn n1 2 3 2 2 1� � � �diag diag… …+ . (14)
Todi z rivnosti (12) vyplyva[ x xi i2 2 1+ + = λ1 + λ2 . Krim toho, x1 = λ1 abo x1 =
= λ2 . Vykorystovugçy tverdΩennq81, otrymu[mo rozklad u linijnu kombinacig
dvox proektoriv dlq koΩnoho dodanka z (14). Takym çynom, rivnist\ (13) vyko-
nu[t\sq.
Teper rozhlqnemo operator D̃ ≈ D :
D̃ = diag ( , , , , , , , )a a a b a xk k k1 2 2 1 2 2 1… …− + .
Todi diahonal\ni elementy riznyci operatoriv
D̃ – X = diag ( , , , )y y y1 2 3 … = : Y
magt\ taku vlastyvist\: y yj j2 1 2− + = 0 i y j ∈ [ λ4 ; λ3 ] , j ∈ N. Zvidsy za
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
PRO ROZKLAD DIAHONAL|NOHO OPERATORA V LINIJNU KOMBINACIG … 393
tverdΩennqm81 matrycq diag( , )y yj j2 1 2− [ linijnog kombinaci[g dvox proekto-
riv z koefici[ntamy λ3 i λ4 . Ce spravedlyvo j dlq prqmo] sumy takyx matryc\:
D̃ – X = Y = λ3 P3 + λ4 P4 ,
D̃ = λ1 P1 + λ2 P2 + λ3 P3 + λ4 P4 .
Lemu82 dovedeno.
ZauvaΩennq$2. Zbil\ßennq znaçennq konstanty c v dovedenni lemy82 ne
vplyva[ na spravedlyvist\ ]] dovedennq.
Teorema$2. Diahonal\nyj samosprqΩenyj operator [ dijsnog linijnog
kombinaci[g çotyr\ox ortoproektoriv.
Dovedennq. Nexaj A = diag ( a1, a2, … , ak , … ) , ai ∈ R. U poslidovnosti a1,
a2, … , ak , … vyberemo monotonnu zbiΩnu pidposlidovnist\ amk
tak, wob
1
∞
→∞∑ −a am k mk k
lim < ∞ .
Rozhlqnemo vypadok, koly amk
[ spadnog. Prypustymo, wo j1, j2, … , js , … —
vsi natural\ni çysla, qki ne vvijßly do pidposlidovnosti mk, k ∈ N. MoΩna
vvaΩaty, wo ]x neskinçenna kil\kist\. Todi za lemog82 dlq koΩnoho s ∈ N ope-
rator
Ds = diag a a a aj m m ms k k ks s s
, , , ,
1 2 3
…
, m
ki
s ∈ Ns ,
[ linijnog kombinaci[g çotyr\ox proektoriv. Bil\ß toho, vnaslidok obmeΩeno-
sti operatora A konstantu c moΩna vzqty [dynog dlq vsix operatoriv Ds
(dyv. zauvaΩennq82). Takym çynom, oskil\ky
�
1
∞
Ds ≈u A ,
to A [ linijnog kombinaci[g çotyr\ox proektoriv, i vypadok, koly amk
spada[,
rozhlqnuto povnistg. Qkwo Ω amk
zrosta[, to poslidovnist\ − am1
, − am2
,
− am3
, … spada[ i tomu – A = λ1 P1 + λ2 P2 + λ3 P3 + λ4 P4 . Zvidsy
A = – λ1 P1 – λ2 P2 – λ3 P3 – λ4 P4 ,
i teorema82 [ spravedlyvog i v c\omu vypadku.
Teoremu dovedeno.
Avtor vyslovlg[ podqku profesoru G. S. Samojlenku za cinni porady i za-
uvaΩennq pry roboti nad ci[g tematykog.
1. Wu P. Y. Additive combinations of special operators // Funct. Anal. and Oper. Theory. – 1994. –
30. – P. 337 – 361.
2. Pearcy C., Topping D. M. Sums of small numbers of idempotents // Mich. Math. J. – 1967. – 14. –
P. 453 – 465.
3. Brown A., Halmos P. R., Pearcy C. Commutators of operators on Hilbert space // Can. J. Math. –
1965. – 17. – P. 695 – 708.
4. Rabanovyç V. I. Pro rozklad operatora v sumu çotyr\ox idempotentiv // Ukr. mat. Ωurn. –
2004. – 56, # 3. – S. 419 – 424.
5. Rabanovich V. Every matrix is a linear combination of three idempotents // Linear Algebra and Its
Appl. – 2004. – 390. – P. 137 – 143.
6. Matsumoto K. Self-adjoint operators as a real sum of 5 projections // Math. Jap. – 1984. – 29. –
P. 291 – 294.
7. Nakamura Y. Any Hermitia matrix is a linear combination of four projections // Linear Algebra
and Its Appl. – 1984. – 61. – P. 133 – 139.
OderΩano 20.04.2004
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
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| id | umjimathkievua-article-3606 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:45:39Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a0/95a9ca3145631567042c1b997b7daba0.pdf |
| spelling | umjimathkievua-article-36062020-03-18T19:59:42Z On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors Про розклад діагонального оператора в лінійну комбінацію ідемпотентів або проекторів Rabanovych, V. I. Рабанович, В. І. For a bounded operator that is not a sum of scalar and compact operators and is similar to a diagonal operator, we prove that it is a linear combination of three idempotents. It is also proved that any self-adjoint diagonal operator is a linear combination of four orthoprojectors with real coefficients. Доведено, що обмежений оператор, який не є сумою скалярного i компактного операторів i подібний до діагонального, є лінійною комбінацією трьох ідемпотентів, а будь-який самоспряжений діагональний оператор є лінійною комбінацією чотирьох ортопроекторів із дійсними коефіцієнтами. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3606 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 388–393 Український математичний журнал; Том 57 № 3 (2005); 388–393 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3606/3943 https://umj.imath.kiev.ua/index.php/umj/article/view/3606/3944 Copyright (c) 2005 Rabanovych V. I. |
| spellingShingle | Rabanovych, V. I. Рабанович, В. І. On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title | On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title_alt | Про розклад діагонального оператора в лінійну комбінацію ідемпотентів або проекторів |
| title_full | On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title_fullStr | On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title_full_unstemmed | On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title_short | On the Decomposition of a Diagonal Operator into a Linear Combination of Idempotents or Projectors |
| title_sort | on the decomposition of a diagonal operator into a linear combination of idempotents or projectors |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3606 |
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