By Joel D. Pincus

Vital currents have been invented to supply a non commutative spectral concept during which there's nonetheless major localization. those currents are frequently fundamental and are linked to a vector box and an integer-valued weight which performs the position of a multi-operator index. The examine of central currents contains scattering concept, new geometry linked to operator algebras, disorder areas linked to Wiener-Hopf and different necessary operators, and the dilation thought of contraction operators. This monograph explores the metric geometry of such currents for a couple of unitary operators and sure linked contraction operators. functions to Toeplitz, singular necessary, and differential operators are integrated.

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**Example text**

Indeed, by (13), (/ — P\)P2 — I — Pi, we have Tx = P2WxP2 = P 2 (Pi e (/ - Pi))M/iP 2 = P2P1W1P2 0 P 2 ( / - Pi)WiP 2 = f i e (/ - Pi)W^i(/ - Pi) = f i © W M . Therefore by taking 7i and T2 as operators on the full Hilbert space H, GT f (C,V) serves also as the principal current for Remark 6: We note that Px = E ™ ^ " 1 Pij {Pi,j>}ti and P 2 = E ^ i + " 2 p2,j- {Ti,T2}. For any subsets C { A , , } ^ ™ and { P 2 j J t i C {P2,>}Jm=21+n2 we can define Qi = J - £ t i *U and Qi — I — HfcLi ft,jfc- Then we may consider the operator four-tuple {Qi,Q2, ^ 1 , ^ 2 } All of the theorems in this section remain true for this four-tuple of operators.

We now show that {WjJ" •} consists of linearly independent vectors. -)) = _ L A ( 0 , r 7+ ) P ( r ? + ) ( l-

Then if p is close enough to 1 we will have f PP(0^ — t)d/i~^(r)) > Saw ( T ^ M I W - Thus -^(ei^AzHre^)) > s i n ^ • ± ± > I ( % ). Accordingly sin%- 1+p ^i(^fc) But sm 2 ! 2 " From the inequality above, we now get the inequality sin ^i±£|Ac>e^)l2 l On the other hand, we have |A(pe^)|2 = ( S e ^ A ^ p e ^ ) ) 2 + ( ^ e ^ A ^ p e ^ ) ) 2 And 9e^Aw(pew*) = sin^/Pp(0- -t)dfi+(rj). Therefore But we know from lemma 24 above that sin^fnt,{rj) = J^V^(T}). Now let p —> 1 and we will get the desired inequality dv+iv) ' 2 ' 2nJ \r, %l 2 < 1 4 1 M% '"' e ' By Theorem 21, we may thus conclude that if we set WT, 3v = ' | ^ .