Материалы к экзамену по функану (1135187), страница 2

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, ϕµ } {ψ1 , ... , ψν }µP´ÛC·iYr±mÊf³kie³h»·i]le¶²i^mi°g_±ljY\kimiYOpÊe_kiY\·ikie[*Y\^ikiecreBe_²iY\^gl]le^gBp A XZeºÒ¹^glki¼iYr]Sx = Ax + (x, ϕj )ψjYrc\e±_mÊfÅÌÝY½f³j=1mikmi·iki¿Rn7[*mik ­ XSpÊe[*²g_pi]lki¿RnϽh]»g_pV·]lep S ²i^im[*Y\kim[*¿¨±XrY¶^glkiY\Y¶²ie_p¹g_q\glkki¿RYC]_Yre ¯^iY[*¿¶´ ¬ e_p¹gl¾dY\[Oh¹·]le Sx = 0 m[*Y\Yr]]le_j®_pÊe]l^im±_mglj®_kieYS^iYrÈdYrkimiY´ kgº·Y Ax = − Pµ (x, ϕj )ψj hj=1b²i^im·iY\[Š²eXZpÊe_j®_p ­ miq²iYr^i±e_nFjY[+[*¿·iY\]l±_Y\^i]le_c\eË°_mijYr]»g˱XZY ψj e^i]le_c\e_kglj®_k¿œêìë A h]_e Ax = 0m (x, ϕj ) = 0 fji¼ j = 1, ..., µ ´ÚùÐ]»g»jeJ°_¿R]_®h x e^i]lecre_k³g»jY\k {ϕ1 , ... , ϕν } m—hÚX½jYZf³e_±gl]lYrj®_kie³hàSárâ A ´)µSe Ax = 0 h#]le3YXZ]l® x ∈ àSárâ A ´ÛC]_XrÆÚfg x = 0 ´ ]»glp)h—àSárâ S = {0}m%²ie3±_]le^ie_n%]lY\e_^iY[*Y8^iY½fcrelj³®[+gih_kglnÊfYr]_XǼ y ·i]_e Sy = ψ ´ÊùÐpÊglji¼i^ikie ­ [*kieº¾m[%o\]le¶^gl±YrkXr]l±_eSkg ψ ´ ¬ e_j ­ ·im[µ+1Þ ¿zf³epÊg_q\gljm²i^e_]lmi±e_^iY\·imiY´ µ+10 = (x, 0) = (x, A∗ ψµ+1 ) = (Ax, ψµ+1 ) = (ψµ+1 , ψµ+1 ) = 1 ¯]l^iY\]l®_ÆU]_Yre_^Y\[ ­ 8^iYZf³cre_j®[+g- ’$d•)˜™;•—¥ t/.

Ÿ v#vt102 ŸC© •—™ t ˜ v ¡;•)– w ™;•œ–O˜¥3¡ t – w ¢C˜«³˜Ù˜¡C•—™ t*w ˜™ t—Y\e_^iY[*3¿ 8^iY½f³c\elj®[+g¶²ieqr±_e_ji¼iÆ;]Bfe_p¹glqgl]l®¸XZjY½f ­ Æ Î8­ Æe_·Yrki®¸²eljY\qrk ­ Ǝ]_Yre_^Y\[ ­ e8XZ²iY\pi]l^iYpÊe[*²g_pi]lkiecreHe_²Yr^g_]le_^gӗY\e_^iY[+gÌÍúm³XrXrg ¯ 8miÈdYr^gSe¸Xr²iYrp]l^iYpÊe[*²³glpi]lke_creSe²iYr^g_]le_^³gϽӻ² ­ XZ]_® K ¯ pÊe[*²g_pi]lki¿RnHe_²Yr^g ¯]le^=±Fcrmij®_°Yr^i]le±_e[Š²i^ieXZ]l^g_kXZ]_±_Y3ÌÍkgFX\g_[*e[Šf³YZj³Y8feXZ]»g_]le_·ike1°glkglÒe_±e_±eXZ]_mV²^ieXr]l^glk³XZ]l±gϽ´—ecÇfg›Xr²iYrpi]_^Še_²iY\^gl]le^g K XZeXZ]lemi]Jmiq7miq\eljm^ie_±glkiki¿OҔXZe_°XZ]l±Yrkik¿OÒqrk³gº·iYrkimn µ pÊekiYr·ike_nkpi^g_]lkieXZ]lmÉÌÍm—hX½jYZf³e_±gl]lYrj®_kie³h³o\]lm¹Ò%qrkgº·iY\kimin=kiYH°_elj³YrY·iY\[œXZ· \]lkieY·mX½jeÊϽh#²i^mi·iY\[ŽY\XZjm%Xre_° ¯XZ]_±_Yrkki¿OÒÃqrk³gº·iYrkimnV°Y\XrpekiYr·kie[*ke_cre³h¹]le µk → 0 ´î8Yrn³XZ]l±mi]lYZj³®_kie³h+² ­ Xr]l®pÊe[*²g_pi]lki¿Rn—4´ —e_cÇfg ∀λ 6= 0 λK ¯ ]leº¾Y3pÊe[*²g_pi]lki¿RnÉmÉpŠYrc\e^iY\qrelj³®_±_Y\ki]lY R−1 = A − λI K²i^im¯ [*Y\kim[*¿É]lY\e_^iY[*5¿ 8^iYZf³cre_j®[+g´_ћ·³g_XZ]_kieXr]lm—h_mqR±_]le^ie_nH]lY\e_^iY[*¿X½j³Y½f ­ Yr]ºhO·i]le”Y\λX½j³m Rλ kYre_°^gl]_m[OhÐ]_e”àSá\â Rλ 6= {0} hOg›miq%]l^iY\]l®_Y\nŽ]lY\e_^iY[*¿ ¯ ·i]leÉàSárâ R−1 ¯pÊe_kiY\·ikie[*Yr^iY\k—h+]º´ Y´+jÆ;°g»¼ÅkYrk ­ jY\±g»¼”]le_·pÊg%XZ²Yrpi]l^³g K YXZ]l®ÂXre_°Xr]l±_Y\kikieYÃqrk³gº·iYrkimYVpÊekiYr·iλke_npi^g_]lkieXZ]lm—´î¸ji¼Žfg»j®kiYrnÈ8Yrc\e²e_]l^iY\° ­ Yr]_XǼ¨X½jYZf ­ Æ Î gl¼u]lY\e_^iY[+giÓ±XǼipiminœpÊe[*²g_pi]lki¿Rn¨e_²Yr^g_]le_^ ±°g_kg»Òe_±e[ÙÌÍm—h+±%·gXZ]lkieXZ]lm)h+±%c\m¹j®°_Yr^]le_±e[϶²^ieXr]l^glk³XZ]l±Y E m[*YrY\] ∀δ > 0 j³miÈ8®%pÊe_kiY\·iAkie_Y·imXZjeÉjmikYrnikieŠkYrq\g_±_mXrm[*¿OҜXre_°XZ]l±Yrkiki¿OÒu±_Yrp]le_^ie±hOe_]_±_Yr·³glÆ Î m¹ÒœXre_°XZ]l±Yrkiki¿[Áqrkgº·Yrkim¹¼[Oh²ieŠ[*ef ­ jÆ ²i^iY\±_eXÇÒef¼ Î m[ δ ´Úî8YrnXr]l±_m]lYZj®kiehO² ­ XZ]l® λ1, ...

, λn , ... ¯ p¹g_pÊgl¼ ¯ j³mi°_e›²eX½j³Y½f³e ¯±g_]lYZj³®_kieXZ]l®%XZe_°XZ]l±Yrkik¿Oқqrkgº·iY\kimine_²iY\^gl]le^g X ­ X½je±_m¹¼[*mhg=Xre_e]l±_Y\]_XZ]l± ­ Æ Î mYm[zXZe_°XZ]l±Yrkik¿RYñ_Y\pi]le^g {xn} jmikiY\nikie%kiYrqgl±_m³XZAm[*¿¶´ ¬S­ Xr]l® En |λ=k |hx>1 , δ..., xn i '´ )e_cÇfg=kgln¹f³Yr] ¯XǼu²eX½j³Y½f³e±gl]_YZj®kieXr]l® yn : kynk = 1, yn ∈ En, ρ(yn, En−1 ) > 1 ´Úî8YrnXr]l±_mi]_YZj®kiehO^g_qjmikYrnikieŠkYrq\g_±_mXrm[*¿¶hR]le xn ∈/ En−1 m ρ(xn, En−1 ) = α >2 0 ´ ¬S­ Xr]l® x∗ ∈ En−1x1 , x2 , ...]»glpÊe_±³h*·i]le ρ(xn − x∗ , En−1 ) < 2α ÌÝX ­Î Y\Xr]l±_e±glkmiYÃe·iYr±mÊf³kieÊϽ´ —ecÇfgDf³eXr]»gl]le·ikie%²ieljeº¾mi]_®e·iYr±mÊf³kie³h)·i]_eD±XrYÃXZ±e_nXr]l±g yn ep¹glqr¿R±glÆC]_XǼ%±¿R²ielj³kiYrki¿[*m)´ÑÚe_q\®[*Y\[œ]lY\²iYr^®−xyn = kxx −xk ¯²ieX½jYZf³e_±gl]lYrj®_kieXZ]l® n y o ¯ X ­ ·iY\]le[ λn > δ ekgecr^g_kimi·iY\kgi´ [*YrY\[ yn = Pn αk xk hXre_e_]_±_Yr] ¯nn∗∗nλn´ ¬ eXrpe_j®_p ­hhBÌ y ∈ E hÏZh]_e8Y\Xr]l®¸miqC²ieXr]l^ieYrkikien²ieXZjY½fe_±g_]lYZj³®_kieXZ]lmHkiYrj®_qr¼Ë±_¿R°_^³gl]l®¸XÇÒefp¼ Î8­ ÆCXÇp¼²ief³²eX½j³Y½f³e±gl]_YZj®kieXr]l® ¯ ²i^e_]lmi±e_^iY\·imiY¶XSpÊe[*²g_pi]lkieXZ]l®Æ ´]»glp—hXZ²iY\pi]l^ [*eº¾Yr]3XZeXZ]leº¼i]l®3]_elj®peFmiq1XZe_°XZ]l±Yrkik¿OÒDqrkgº·YrkiminJpÊe_kYr·ikienJpi^gl]_kieXr]lmÂmk ­ ji¼—hgSmiqR²i^iYZf³¿Úf ­ÊÎ Y\n]_Yre_^Y\[*¿”jYrc\pÊeB²elj ­ ·imi]_®¶me_° Î YrY ­ ]l±_Y\^¾dfYrkimiYXr²iYrp]l^g»j³®_kien]_Yre_^Y\[*¿úmX\Xrg ¯ 8miÈdYr^gÌÍmiq\eljmi^e_±g_kikieXZ]l®DjÆ;°e_nÅkYrk ­ jY\±_e_nÅ]le_·ipm”Xr²iYrpi]_^gJX½jYZf ­ Y\]DmiqV]le_c\eh*·i]le%or]_ee_q\kgº·gljed°¿uk³g»jmi·miY¶°_YXZpÊe_kiY\·ikiecre·imX½jgXZe_°XZ]l±Yrkik¿OÒÃqrkgº·Yrkimin)hÊf³eXr]»gl]le·ikie°lj³miqrpimiÒ1pFor]len]le·ipÊYBm)hiXZ]»g»jeH°¿R]l®h¹°Y\XZpÊekiYr·ike_cred·imXZj³gdXre_°XZ]l±Yrkiki¿OÒËq\kgº·iY\kimin—h¹°elj®È8m¹Ò˲ie[*ef ­ jƨkiY\pÊe_]le ¯^iecreòeljeº¾mi]lYrj®_ke_cre δ Ͻ´)ùœk ­ jY[Ohp%XZeº¾Hg»jY\kimiÆ8hf³Yrj³gÃe°Xr]leº¼i]ËkiYXZ]le_j®ÃÒe_^ieÈ8e³´³ÑÚe ¯ ²iYr^±_¿OÒ#hXZ²Yrpi]l^Dpe[*²glp]lkie_c\eËe_²Yr^g_]le_^g˱Ëcrm¹j³®_°_Y\^i]le±_e[²i^ieXZ]l^³glkXr]l±_Yd±XZY\cÇfgXZefYr^¾mi]Ë]le·ip ­ aih³Òe_]¼e_k³gRm¸kiYÐe_°º¼iq\g_kgÚ°_¿R]l®BXZe°XZ]_±_Yrkki¿[Vqrkgº·iY\kimiY[ K ´Zî8Y\nXZ]_±_mi]lYrj®_kehr² ­ XZ]_® K −1 e_c\^glkmi·iYrk)hr]lecÇfgY\XZ]_®²i^ie_mqr±_YZf³YrkmiY¶pÊe[*²g_pi]lkiecreHkgHe_c\^glkmi·iYrkki¿Rn pÊe[*²g_pi]lki¿Rn3e²iYr^³gl]le^ ¯ keI = K · K −1±Ãc\m¹j®°_Yr^]le_±e[¨ÌÍ°Y\XZpÊekiYr·ike[*Yr^kie[ϲi^ieXr]l^g_kXZ]l±Y I kiYdpÊe[*²g_pi⇒]lYrk)´Øz^g_q −1 kYre_c\^glkmi·iYrk)h]le K = K − 0 · I ¯ kYre_°^gl]_m[Oh]º´ Y_´ 0 ∈ σ(K) ´µSe_j®¸kiYCe_°º¼iq\g_kH°_¿R]l®¸fg»¾Ymiqre_jKmi^ie±g_kikie_nH]le·ipÊe_nXZ²Yrpi]l^³g kg_²i^ie]lmi±h]lYre^iY\[+g ­ ]_±_Yr^¾fg_Yr]ºh_·i]_e¸e_k° ­ fYr]¶kiY\miqre_jmi^ie±glkHYX½jm K ° ­ fYr]8Xref³Yr^¾g_]l®°_YXZpÊe_kYr·i¯ kieYC·mX½jeXre_°Xr]l±_Y\kiki¿OÒÃq\kgº·iY\kimin—´XZ]_±_Yrkkien−1PhicÇf³Y+ α n xn = y n + z nk=1n−1Pα k λkA λynn =zn =αk λλnk − 1 xk ∈ En−1λnk=1 k=1 yy∀p > q A λpp − A λqq = kyp + zp − (yq + zq )k = kyp − (yq − zp + zq )k > 21yq − zp + zq ∈ Ep−1AA6­ 7 m¹j®°_Yr^]ºg [*mÊf³]»gh²e_p¹glqglkik ­ Æz±1²i^e_ȸj³e[XrY\[*Y#g[*Yr·g_kimiY_Ókg_²ie[*kim³[q\g_ef³kie1]lYre^iY\[ 8ZX ]_^iY_ÓÖfji¼ŠjÆC°_ecreÂpÊe[*²g_pi]lkiecreÅX\g_[*eXZe_²^¼¹¾Yrkike_cre%j:¯ m9 kiYrnike_creÅe_²iY\^gl]le^g A ±crm¹j³®_°_Y\^i]le±_e[ ¯²i^ieXZ]l^³glkXr]l±_Y H X ­ÊÎ Y\Xr]l± ­ Yr]¶e^i]le_c\e_kglj®_k³g»¼dke_^[*mi^e_±g_kikgl¼HXrmXZ]lY[+g {ϕn } XZe_°XZ]l±Yrkik¿OÒd±_Y\pi]le ¯^ie±h)Xre_e_]_±_Yr]XZ]l± ­ Æ Î m¹ÒJXZe°XZ]_±_Yrkki¿[q\kgº·iYrkm¹¼[ {λn } ÌÝX ­ ·iY˲^imi·iY[uYX½jmÂo\]»g3XZmXr]lY\[+g3°Y\Xrpe ¯kiY\·ikgih]le;´ )Yre_^Y\[+g 7 m¹j®°_Y\^i]»g [*mÊf]ºg8°_e_jYrY;XZj³gl°g»¼—hkiYZ¾YZjmÃ]lYre^iY\[+gdúmX\Xrg 8miÈ8Y\^gih²ieXZpÊelj®p ­ λn]l^i→Yr° ­ 0Y\]=Xrg[*eXre_²i^¼¹¾dY\kikieXZ]l<¯ 9 m›e²iYr^³gl]le^gÅÌôgDX ­ÊÎ Y\XZ]_±_e_±glkimYËe_^i]_e_crekg»j³®_kie¯cre7°glqrm³Xrgfji¼3Xrg[*eXre_²i^¼¹¾dY\kikiecree_²iY\^gl]le^gih¹±H²i^mikiÔimi²Y_h³[*¿¨²iep¹glqr¿R±g»jm1e_]\f³Yrj®_kien1]_Yre_^Y\[*e_n ¯ Òe_]¼1]º´7 ¯<9 f³ep¹glqr¿R±g»j³gXZ®¸²e ¯ f³^ ­ cre[ ­ ÏZ´=O’>Ž™CŸ w •—™;Ÿ? w ˜«³˜ w ˜z–O˜¥3¡ªB•—– v ¢;˜•@;Ÿ v ªC˜Ù¡;™CŸ;¢ t ÿSªC•#þğ wzv ¡;•)– w ™A¢;˜™C¥ t ª x ¢;˜Ö«³˜Ä˜¡C•—™ t*w ˜™ t ’SY\Xr]lkie7c\e_±e_^¼—h)o\]le_]=°_m¹jY\] ¯ kg_Xr]leº¼ Î mnÂ]lY\[*ki¿RnjYXlh²¹jeºÒeDe_²imX\glkik¿Rn›±7jY\piÔim¹¼¹ÒÂm›fji¼pÊe_]le^ie_c\eF]\¼¹¾YZje3k³glni]lm%Òe_]l®3·i]_e ¯ ]le3±Fjmi]lY\^gl] ­ ^Y_´#î¸j¼%kgº·³g»j³gV^g_qr°_Y\^iY\[+XǼJXH]lY\[Oh·i]le3]»g_peYkie^[+g»j®ki¿Rn8e²iYr^³gl]le^—´ Þ kiYOcre±_e^i¹m jm—hº·i]_eBo\]leSe_²iY\^gl]_e_^—hºpÊe[+[ ­ ]lmi^ ­ Æ Î minHXZe¶Xr±_e_m³[7XZe²i^¼¹¾dYrk ¯ki¿[Oh]º´ Y´ A : AA∗ = A∗ A h\kie;]ºg_p¶or]leCm¹jm¸kYr] ­ qrk³gl]l®±jmi]lY\^gl] ­ ^iYÐkiYÖ²elj ­ ·im¹j³eXZ®³´rÑ%·g_Xr]lkieXZ]lm—hY\XZjm A ¯ Xrg[*eXre_²i^¼¹¾dY\kiki¿RnËmijm1o\^[*mi]le±_¿Rn—hÊ]leHekÃkie^[+g»j®ki¿Rn—´%S]leq\gdq\glcZg»f³e_·ki¿RnËpi^im]lYr^imn±¸o\]le[°m¹jY\]lY ­ ²ie[*mkglY\]_XǼ ¯ kiY\miqr±Y\Xr]lkiehÊXèfg_±_È8mYÚf³eXr^ie_·ike¸²i^iYZf³²ieljglcZg»j³m—h_·i]_e8m[*Y\Yr]_XǼ˱±_mÊf ­Xrg[*eXZe²i^¼¹¾dY\kiki¿Rn7ÌͱYr^ieº¼i]_kiehq½f³YXZ®¶feXZ]»g_]le_·ike¸kie_^—Y\e_^iY[+gÃÌÝpi^imi]lY\^iminVÑÐY\n¹ji¼³ÏZÓ² ­ XZ]_®[+g»j³®_kieXZ]lm³ÏÖe_²Yr^g_]le_^˱dpÊe[*²¹jY\p¹XZkieqrkgºA·ik¯ e[Jcrmij®_°Yr^i]le±_e[=²^ieXr]l^glk³XZ]l±Y H hÊ]lecÇfg λ ∈ σ(A) ⇔¯´∃xn : kxn k = 1, k(A − λI)xn k → 0î8e_p¹g_q\gl]_YZj®XZ]l±eÓ ⇐ Y\X½j³m”X ­Î Y\Xr]l± ­ Æ;] ­ p¹glqglkiki¿RY xn h+]le (A − λI) kY7[*eº¾dYr]%m[*Y\]l®Âe_c\^g ¯kimi·Yrkikiecre3e°_^g_]lkie_c\eh—²eXZpÊe_j®_p ­ ±7or]le[¨X½j ­ ·g_Y yn = (A − λI)xn → 0 jÆ;°e_nJe_c\^glkim·iYrkik¿RnÌ kiY\²i^iYr^¿R±_ki¿Rn³ÏCe²iYr^g_]le_^7f³elj¾dYrk%°_¿Oj%°_¿²Yr^iY\±_Y\Xr]lm%C± BD(÷ë n = ABD(÷ë yn = A0 = 0 hkieF²i^im]l⇒e[ BD( ë (A − λI)−1 yn = BE( ë (A − λI)−1 yn = lim kxn k = 1 Ay¯ ²i^ie]lmi;´ ±—e_e_^icÇY\f·ig¸miY_àS´ árâ­¬²ZX_]®´i^ZY³fi²_ejºe¾mO[lhi·l]e´ùÐ]»g»⇒je°_¿R]l®³hλ λ∈ kiσ(A)Y=jYr¾Hm]±›f³mXZp^iYr]lke[ÁαXr²i=Yrpinf]l^iY_kxk=1´ ¬ ^iYZkAf³²i−e_jλIkeº¾m>[OhÐ0·]leekie›j³YZ¾(Ami−]ÅλI)±”XZ²i=Y\pi{0}]l^iYeXr]»gl]le·ikie[O´+ÑÄo\]le[ïX½j ­ ·³glY ∃h 6= 0 : ∀x ((A − λI)x, h) = 0 ´ ¬ Yr^iY\²imXZ¿R±g»¼X ­ ·iY\]le[zXrg_[*e ¯XZe²i^¼¹¾dY\kikieXr]lm A h ∀x (x, (A − λI)h) = 0 ´+ÛC·iYr±mÊf³kie³h (A − λI)h = 0 ⇒ λ ∈ σp A ´ù ­ ·iYr]_e[Xrg[*eXre_²i^¼¹¾dY\kikieXZ]lm A h³±XZYSY\creËXZe°Xr]l±_Y\kiki¿RYSqrkgº·Yrkim¹¼F²i^imkgºfj³YZ¾g_] R Ó³±_YZf³® λ khk = λ(h, h) =´»ùÖjY½f³e±g_]lYZj®kieh·iYrc\ehrp¹g_p¸[*¿ ­ ¾Y²ie_p¹g(Ah, h) = (h, Ah) = (Ah, h) = λ khk ⇒∈ λ ∈ Rq\gljm—h»°¿R]l®CkY[*eº¾dYr]º´ÛBXr]ºg_Yr]_XǼd²ieXZjY½f³kminXZj ­ ·gln ¯ kiY\²i^iYr^¿R±_kieλcr∈eCXrσ²ipYrAp]l^giÓ (A−λI)H ²¹j³e_]lkieS± ¯h»kieSkiYO±FX H !´ —ecÇfg ∃y ∈ H \(A−λI)H : y 6= 0 h»±¶XZm¹j ­ ²¹je_]_kieXr]lm ∃xn ∈ H : (A−λI)xn → y ´HkiYr²i^Yr^i¿R±Yrk—hC²i^imi·iY[¤²ie ­ XZje_±miÆ k(A − λI)xn − (A − λI)xk k ≥ α kxn − xk k ´Cµ ­ g(A − λI)À ­k²i^im±›XZm¹j ­ ±_¿R°e_^g ´OùÐ]»glje°_¿R]l®³hk(A − λI)xn − (A − λI)xk k → 0fg[*Yrki]»g»j³®_kgl¼=²ieX½jYZf³e_±gl]lYrj®_kieXZ]l® ⇒n,∃xm0 =→ BD(÷∞ë xn m²ieFkiY\²i^iY\^i¿Rx±_kineXZ]lm (A − λI)x0xn=¯ y h—]º´ Y_¯ ´²i^ie_]_mi±_e^iYr·imY¸XS²i^iY½f²ieljeº¾YrkimY\[O´ùÐ]»g»j³e°_¿R]l®³hior]leÃm3kiY¶]le_·pÊgHkYr²i^iY\^i¿R±_ke_crey ∈ (A − λI)H ¯XZ²Yrpi]l^³gHmV²e_]le[ ­ λ ∈/ σ(A) ¯ ²^ie_]lm±_e_^Yr·imiY´ —Y\e_^iY[+gdf³e_p¹glqglkgGO’$d•)˜™;•—¥ t ˜ï˜ w ˜™ t þ •)¢;Ÿ;Ÿ v ¡C•—– w ™ t—Y\e_^iY[+giÓº² ­ XZ]l®pÊe[*²¹jYrp¹Xrkie_q\kgº·ikieY°g_kg»Òe_±_eO²i^eXZ]_^glkXr]l±_e³h : X → X ¯ kYr²i^iY\^i¿R±_k¿Rnj mikYrniki¿Rn=e_²iY\^gl]_e_^—X´*—¯ ecÇfg ∀[*kie_c\e_·¹jY\kg P (z) m[*YrY[ σ(P (A)) = P A(σ(A))´)µ¶gHjY\piÔim¹¼¹Ò7o\]le1kYf³epÊg_qr¿R±g»jeXZ®h+ke›]lYre^iY\[ ­ [*eº¾kie ­ XZmijmi]l®ÅXZjY½f ­ Æ Î m[Ùe°_^g_qre[OÓ ∀ ^g_Ôimiekg»j®kie_nŠÀ ­ kipiÔmimh¹kiY¶m[*Y\Æ Î YrnF²ielj³ÆCXZe±Hkgm³[*YrY\[´R(z)= R(σ(A))î8e_p¹g_q\gl]_YZj®XZ]l±eÓ² ­ Xr]l® r(x)σA= p(x)q(x)σ(R(A))ÝÌ\Y½X³jÅmf_e¹plg\qR¿±glY[ XZj³gl°¿Rn±gl^img_ki]ºh]le²i^eXZ]_e−1ÜZÏhm±\q_gm*[ikei²i^eZXl]R¿dY*[ik_e\c_e¹·j\Yik¶¿´½XjmpÊe_^ikim p ÌÝX ­ ·iYr]le[Épi^g_]lkie ¯¯q(x) = 1 p q ¯α1 , ..., αm ¯mnQXZ]_m H ÏZh β1, ..., βm ¯ pÊe_^kim q h]le r(x) = c Qm (x − αi ) Qn (x − βj )−1 m r(A) = c Q(A − αi I)(A −i=1j=1i=1j=1W¯ Xe_°ljg_XZ]_®_Æïe_²i^iYZf³YZj³Yrkim¹¼%±Xdkie^[+g»j³®_kieJÌÍkg_²ie[*km[Oh—·i]le^Yrqre_ji=1®_±Yrki]»gde_²iY\^j=1gl]_e_^g ¯ kiYr²^iYr^i¿R±kg±]le_·pÊglÒ#h¹kiYBjYZ¾Hg Î miÒñXZ²iY\pi]l^iYºÏ½´Rλ A ¯¬S­ Xr]l® λ ∈ σ(A) hOg µ = r(λ)´Oú+glÔmie_kglj®_k³g»¼”À ­ kipÔim¹¼ r(x) − µ e_°^g Î glY\]_XǼ±Âk ­ j®Å²i^imÊhYCi·m½X³jiml]ZY³jd®m*[\YrY]Êpei^ikY\[Jm)hÊXZjY½fe_±g_]lYZj³®_kie³hXZefYr^¾mi]8[*kieº¾mi]lYrj®´ ¬ e_or]_e[ ­x=λe_²Yr^g_]le_^ r(A) − µI Xref³Yr^¾λ mi]=Xr±_em[[*kieº¾mi]lYrjY\[ kiYre°_^g_]lm[*¿Rn”e_²iY\^gl]_e_^ A(x−−λIλ) m²e_]le[ ­kiY\e_°_^³gl]lm[Oh¹]º´ Y_´ r(λ) ∈ σ(r(A)) ´ÑŽef³k ­ XZ]_e_^iek ­ ²iep¹glq\gljm—´¬S­ Xr]l® µ ∈ σ(r(A)) ´ ¬ ^iYZfXZ]»g_±_m[¨^g_Ôimiekg»j®k ­ ÆðÀ ­ kipiÔmiÆ r(x) − µ ±%±_m¹f³Y1²^ie_miq\±_YZf³Yrkimi¼nmQQ´ Ü XZjm”°_¿@±XZY αi ∈ ρ(A) h+]leJe_²iY\^gl]le^ r(A) − µI m[*YZjŠ°_¿(x − βj )−1r(x) = c (x − αi )βj I)−1 = cmQ(A − αi I)nQRβj (A)e_°^gl]_ki¿RnFe_²iY\^gl]le^ c−1 Qm Rα Qn (A − βj I) h·i]le³he·iYr±mÊf³kie³hikiYr±Yr^ike´ùÐ]»g»j³e°_¿R]l® ∃αi ∈ σ(A) ´µSe r(αi ) − µ = 0 ⇒ µ =i=1r(αi ) j=1´ ¬ epÊg_q\gljmD±_jeº¾dY\kim¹¼± σ(r(A)) m%e°_^g_]lkie³hXZ]»gljeH°_¿R]l®³h¹±_¿R²ie_jkiYrkedmXrpÊe[*e_YS∈^g_r(σ(A))±_Y\kXZ]l±ehÊ·i]lek³g_[mV]_^iYr°e_±gljer(σ(A))Xr®´i=1j=1i‘IR’5J w ˜Ö™ t þ •—¢CŸ;•¨¡;™C˜ vw ™ t ¢ vw ý t ;¢ •—¡C™;•—™Cš›ý¢;šLKNMOO¢C–PCŸ×¢ tïv ¡;•—– w ™;•vt ¥3˜ v ˜¡;™Ú£Oþ •—¢C¢;˜«˜Ù˜¡;•)™ t*w ˜™ t ýï¡C™;˜ vw ™ t ¢ vw ý˜L(H)Õ Y\[+[+g›ß_Ó)² ­ XZ]_® A ¯ Xrg_[*eXZe²i^¼¹¾dYrkki¿Rn—h Q (x) = (Ax, x) ¯ Yrc\e7pi±gºf³^g_]lmi·ik³g»¼›ÀSe_^[+g´Q)e ¯cÇfg sup´î8Y\nXZ]l±mi]lYrj®_kie³heA·iYr±mÊf³kg7e_ÔiY\kip¹g suph*ef³kg_pÊeQQA (x) ≤ kAkA (x) = kAkkxk=1[*eº¾kie3²ie_j ­ ·imi]l®7mÂe°_^g_]lk ­ Æ8Ó²i^ie±_Yr^¼iY\]_XǼ—h—·i]le QA(x + y) − QA(xkxk=1áh)fg»jY\Y− y) = 4R (Ax, y) Y\XZjm suphO]le 4R á (Ax, y) ≤ c kx + yk2 + kx − yk2 ≤ 2c kxk2 + kyk2 ²ekxk=1 |QA (x)| = ckiY\^gl±YrkXr]l± ­ ²gl^gljijYrje_c\^g_[+g´ ¬ elj³g_cZg»¼ y = kxk Ax ²ie_j ­ ·im[ 4 kxk kAxk ≤ 4c kxk2 ¯ ±_e]ÂmkAxk²i^ie]lmi±e_²ie_jeº¾kg»¼ÃeÔiYrkpÊg´Õ Y\[+[+g¸`i´»î¸ji¼ËX\g_[*eXre_²i^¼¹¾YrkikiecreSe_²Yr^g_]le_^g A r(A) = kAk ÌôXZ²iY\pi]l^glj®_k¿Rn^gºf³m ­ X\ÏZ´ ¬ ^ie Î Y±XrYrc\e7Y 1²ie_p¹glq\¿R±g_]l®h—YX½jműXZ²e[*kimi]_®D]lY\e_^iY[ ­ eD]le[¨·i]le r(A) = lim pkAnk hk ­ g kAn k2 = Ì jY[+[+gßÏm)hX½j³Y½f³e±gl]_YZj®kieh 2 A = kAk2 hsupkxk=1 (An x, An x) = SUTV kxk=1 (A2n x, x) = A2n fglj®_ÈdYB±FX Se_·Yr±_m¹f³kie´Õ Y\[+[+gHí´ ¬S­ Xr]l® ¯ Xrg_[*eXZe²i^¼¹¾dYrkki¿Rn1±Hc\m¹j®_°Yr^i]_e_±_e[²i^ieXZ]l^³glkXr]l±_Y H h p ¯ [*ke_cre·¹jY\k1kg´C )e_cÇfg kP (A)k = AmaxQ´ î8Y\nXZ]_±_mi]lYrj®_keh—² ­ XZ]l® p(z) = P cnzn ´|P (t)| ≤ maxt≤kAk |P (t)|P—ecÇfg p(A)∗ p(A) = P c t∈σ(A)#h_ei²\Y^lgl]e^ Q(A) e·iYr±mÊf³kie1Xrg_[*eXZe²i^¼¹¾dYrkki¿Rn—´kck Ak = Q(A)kAîdg»j³YrY_h kp(A)k2 = sup (p(A)x, p(A)x) = sup (p(A)∗ p(A)x, x) = sup (Q(A)x, x) =kxk=1 ´ÌÍm³XZ²—´+²iY\^i± ­ Ƥkxk=1jY[+[ ­ Ïdm”²ieJ±_]le^ie_nÅjY\[+[*kxk=1Yh = r(Q(A)) = supkQ(A)kλ∈σ(Q(A)) |λ| σ(Q(A)) =ÌͲ^iY½f´°_m¹j³Yr]ÏZh]»glp·]le kp(A)k2 = sup p(A)p(A) he_]lp ­ fg7e_·iY\±_mÊf³ke ­ ]_±_Yr^¾f³Y\kimiYQ(σ(A))λ∈σAjY[+[*¿¶´—Y\e_^iY[+g¸ÌÍk³glpÊe_kiY\Ô ¯ ]le¸Ó ¯ ÏèϽ´ ¬S­ XZ]l® A ¯ X\g_[*eXZe_²^¼¹¾Yrkik¿RnBe²iYr^³gl]le^HÌݱ;c\m¹j®_°Yr^i]_e_±_e[˲i^eXZ]_^glk ¯XZ]_±_YÏZh#]le_cÇfg c\e[*e[*e_^ÀSmiq[ gljcrY\°_^i¿ÌÝgljcrY\°_^i¿zkiYr²i^Yr^i¿R±ki¿OÒDÀ ­ kipiÔimin%kg ÏC±²i^ieXZ]l^³glkXr]l±_e ∃!L(H) kiYr²i^Yr^i¿R±Jki¿OқjmikYrnikiC(σ(A))¿OÒe_²iY\^gl]le^ie_±%kg H XZeJX½jYZf ­ Æ Î m³[*mÉXr±_enXZ]lσA±g_[*m—Ó±_e ¯ ²Yr^i±¿OÒhfj¼1jÆC°_ecreH[*kie_c\e_·¹jY\kg p J(p) = p(A) h±e ¯ ±_]le^i¿OÒ#h kJ(f )k = sup´|f (A)|î8e_p¹g_q\gl]_YZj®XZ]l±eÓ² ­ Xr]l® f ∈ C(σ(A)) ´ÚÑÚeqr®[*Y[z[*ke_cre·¹jY\ki¿ Pj ⇒ f kg t∈σ(A)´ ¬ e=j³Y\[+[*Yσ(A)­íih kPj (A) − Pi (A)k = k(Pj − Pi ) (A)k = max²eZXÊp_ej_®pkgPj ⇒ ft∈σ(A) |(Pj − Pi )(t)| → 0 ¯ ­ ­¬dέ´lejºe¾m[´Sãei^i^\Yipl]ikeZXl]7®ÝÌ\Y½X³j3m_±R¿_°\Yi^\Y›[³f^c Æi²i^im_°j¹m¾_gÆÆσ(A)²ieX½jYZf³e_±gl]lYrj®_kieJ(fXZ]l® )P=j é limmÊfj→∞YZ¼ ¯ eP°_jö—(A)YZf³mikimi]_®Ãe_°Yd²eX½j³Y½f³e±gl]_YZj®kieXr]lm7±Vef³k ­ ϽhkiY\²i^iYr^¿R±_kieXZ]l®³h±_¿R²eljkiY\kimiY=²iYr^i±e_c\eÅmŽ±]le_^e_cre ­ XZje_±min—hÚmikiö—Y\pi]lmi±kieXr]l® j³YrcrpÊe²i^ie_±Yr^¼iÆ;]XǼ ÌͲie±_]le^iki¿[²i^im³[*YrkiY\kimiY\[ujY[+[*¿ÄíϽ´ Ü f³mikXr]l±_Y\kikieXZ]l® J X½jYZf ­ Y\]FmiqË]lecrJeh)·i]le3kg3[*kieº¾Y\Xr]l±_Y[*kiecre_·ijYrkie±±XrY¸Y Säñ±e_q[*eº¾Hk¿RY¸^iY\gljmiqglÔimim¹æ¸Xre_±²gºfg_Æ;]ºh³gËor]leV[*kieº¾dYXZ]l±e˱XrÆÚf ­ ²¹je_]_kieñ C(σ(A)) hfgljYrY²ie_j®_q ­ Y\[+XǼ1kiYr²i^Yr^i¿R±kieXr]l®_Æ J ´ùÖjYZfXZ]l±miYVß_ÓY\XZjm kYr²i^iY\^i¿R±_k³gÃkghh]_eh]leVY\XZ]_® ∀x (f (A)x, x) ≥ 0ÌÍe·iYr±mÊf³kgl¼1²iefXZ]»glkie±_p¹fg¸±H²i^Y½f³¿Úf ­ÊÎ YrYCf³σ(A)e_p¹g_q\glf]_YZ≥j®0XZ]l±e ff ≥(A)0ÏZ≥´ 0nnnWÖù jYZfXZ]l±miY1`iӗY\XZjmÅXrg[*eXre_²i^¼¹¾dY\kiki¿Rn A ≥ 0 h)]º´ Y_´ ∀x (Ax, x) ≥ 0 h]le ∃ X\g_[*eXre_²i^¼¹¾Yrkiki¿Rn]ºg_pÊe_n—h#·i]_e 2 ´#î8YrnXr]l±_mi]_YZj®kiehh)À ­ kipiÔim¹¼ f (x) = √x k³g [0, kAk]B ≥0∈ [0, kAk]kiY\²i^iYr^¿R±_kgh¹]ºg_p1·B]lee=Xr]ºg_AYr]_XǼV²ielj³eº¾Hm]l® B =σ(A)Í́±rX*[¿ZXjY B = JA(f )ÏZ´f (A)‘‘+’$•—˜™;•)¥ t ˜ïŸ§—˜¥3˜Ö™?MuŸ§—¥3• vt ¥3˜ v ˜¡;™Ú£Oþ •—¢C¢;˜«˜Ù˜¡;•—™ *t w ˜Ö™ tzvP;ŸC–ªC?Ÿ ;• v –CŸ;¥ ý•—– w ˜™;˜¥ Ÿ×˜¡;•)™ t*w ˜™ t O¥3¢C˜)þz•—¢;Ÿ;£Á¢ tzt ™C«XR¥3•—¢ w ý¡;™C˜ vw ™ t ¢ vw ý•@M5R¢C–P;Ÿ?×¢ t •—«³˜ v ¡C•—– w ™;• –;ý t ÿ¶™ t*w Ÿ;¢C˜ 0 ŸC¢ w •—«™;ŸC™A֕—¥3šLK ¡C˜Ù¢;•—–O˜ w ˜™C˜ ¥3•—™;•’ÛC²i^iYZf³YrkmiY_ÓlÑÚYrp]le_^ h kg_qr¿R±glYr]XǼ8Ôimip¹j³mi·iY\Xrpim[Dfji¼He_²Yr^g_]le_^g A h_Y\X½j³m²eX½j³Y½f³e±gl]_YZj®kieXr]l® m[*Y\Yr]H±XrÆÚf ­ ²¹je]lk ­ ƨjmikYrnik ­ Æe°_e_je_·ip ­ ´h, Ah, A2 h, ...¬ ^im[*Y\^Uß_Óв ­ Xr]l® A ¯ XZm[+[*Y\]l^imi·kg»¼Ž[+g_]l^imiÔg´Y)e_cÇfg A kiY7m[*Y\Yr]ÅXre_°XZ]l±Yrkiki¿OҔ·imXrYZj ⇔m[*Y\Yr]ÂÔimp¹jmi·iYXZpimnŠ±_Yrp]le_^—´ÐÑ@·gXZ]lkieXZ]lm)hÖY½fmikimi·ik¿RnŠe_²iY\^gl]le^ɱ°_elj³YrYF·iY\[ïef³ke[*Yr^kie[A²i^ieXZ]l^³glkXr]l±_YSkiY¶m[*Y\Yr]HÔimipijmi·iYXZpim¹ÒF±_Y\pi]le^ie_±³´¬ ^im[*Y\^F`iÓ H = L2[0, 1]; Ax(t) = tx(t) ´³ÑÚYrpi]_e_^ h(t) = t ¯ Ôimip¹jm·iY\Xrpimin1fji¼ A ´ÛC²i^iYZf³YZj³YrkimiYÓcrmij®_°Yr^i]le±_¿¶h m A2 ¯ e_²iY\^gl]le^i¿Â±;km¹Ò#´»óÐ]lmde_²iY\^gl]_e_^i¿Âkg_qr¿R±glÆ;] ¯­ kmi]»gl^iki¿RnFmiq\e[*e^ÀSmiq\[XXǼ ­ kmi]»gl^ikieHo\pi±_miH±g»1j,Y\Hki]l2ki¯ ¿[*m)hiY\XZjm ∃J : AH1 1 →´−1Z kmi]»gl^iki¿[umiki±gl^im³glki]le[kg_qr¿R±glYr]XǼD]leh#·]le3kiHYH2miq\¯ [*Y\k¹¼iYr]XǼ%²i^m ­ kimi]»g_^iki¿OÒDmiq\Ae2[*=e^JÀSmiq\A[+1glJÒ#´ÑŠ·gXZ]lkeXZ]_m—hXZ²Yrpi]l^)hkie^[+gih¹kgljmi·imY¶Ôimip¹jm·iY\Xrpim¹Ò1±Yrpi]_e_^ie± ¯ ­ kimi]»g_^iki¿RY¶miki±gl^im³glki]l¿¶´—Y\e_^iY[+gi´ ¬S­ Xr]l®Xrg[*eXre_²i^¼¹¾dY\kiki¿RnSe_²Yr^g_]le_^¶±Ccrm¹j®°_Y\^i]le_±e[1²i^ieXZ]l^g_kXZ]_±_Y H hrm[*Y\Æ Î mnÔimipijmi·iYXZpimin”±_Y\pi]leA^—[´ ¯ )e_cÇfg ∃ ]»g_pÊgl¼›kiYre]l^imiÔ³gl]lYrj®_kgl¼›°_e^iYZjY\±XrpÊgl¼›[*Yr^g µ kg σ(A)ÌÍmijmÅkg­­ZÏihi·l]eikim»]lg^ikHeroip±im±lgjrYik_]rYVk_ei²\Y^lgl]e^±´[− kAk , kAk]= tx(t) L2 (µ)î8e_p¹g_q\gl]_YZj®XZ]l±e´A¬S­ XZ]l® f ∈ C(σ(A)) ´ ¬ e_jeº¾m[ l(f ) =Λx(t)h#cÇf³Y h ¯ ]le]VX\g_[*¿Rn=Ôimip ¯jmi·Y\XZpmin1±Yrpi]_e_^—´ÛC·iY\±_mÊf³ke l ¯ kiYr²i^Yr^i¿R±ki¿RnÃj³mikiYrnki¿RnVÀ ­ (fkip(A)h,Ôimie_k³g»h)j1k³g C(σ(A))´ Ü XZjmh]le f (A) ≥ 0 ´ ¬ e]lY\e_^iY[*YBúÖmXrX\g ∃ kiY\e_]l^imÔgl]lYrj®_k³g»¼Ã°e_^iYrjYr±XZp¹g»¼Ã[*Y\^g µ : l(f ) = R f & µf ∀f≥ 0∈σ(A)\´ glpF·i]le (f (A)h, h) = R f & µ ´C(σ(A))N¬ elj³eº¾Hm³[ U (Ak h) = P h)cÇf³Y P (t) = tk ´*ÛC·Yr±_m¹f³kie U : H → L2(µ) ´ Ü X½jm x = Phck Ak hkkN]le U (x) = P´ ¬ eXr[*e_]_^im[Ù·i]le¾dY U q\gÅe_]le°_^gl¾dY\kimiY_Óб_Y\pi]le_^ ­c k Pke_kUXr]ºg_±_mi]”±iXZee_]l±Yr]_Xr]l±_mYJ[*ke_cre·¹jY\k px miq L2(µ) hC²i^imi·iY[ (U x, U y)R PRPP&&( ak tk bk tk ) µ = ai bj ti+jix ∈=Rh, Ah, A2 h, ...&px (t)py (t) µ =σ(A)PPPPµ = ai bj (Ai+j h, h) = ai bj (Ai h, Aj h) = ( ai Ai h,bj Aj h) =#h ]º´ Y_´ U XZeºÒ¹^glki¼iYr]Ãi,jXrpÊglji¼i^ikieYd²^ie_miq\i,j±_YZf³YrkimY_´—Ûf³kie_q\kgº·ikii,jeXZ]l® U e_·Yr±_m¹f³kgihmiqHiXZeºÒ¹^g_kiYrkm¹j ¼XZp¹glji¼i^ikiecre8²^ie_miq\±_YZf³Yrkimi¼1X½jYZf ­ Y\]dmVkiY\²i^iYr^¿R±_kieXZ]l® ±de° Î Y[OhÊor]ledf³Y\nXZ]l±mi]lYrj®_kie ­ kimi]»gl^ki¿Rnmiq\e[*e_^³ÀSmiq\[ h, Ah, A2h, ...

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