pol=1-((2 (((-1+T))^2) (1+(T^4)) ϵ)/(T^3))+(1/(T^6))2 (((-1+T))^2) (6-15 T+12 (T^2)+2 (T^3)-3 (T^4)-2 (T^5)-3 (T^6)+2 (T^7)+12 (T^8)-15 (T^9)+6 (T^10)) (ϵ^2)-(1/(3 (T^9)))(ϵ^3) (360-2520 T+7632 (T^2)-12510 (T^3)+10899 (T^4)-2988 (T^5)-2942 (T^6)+2731 (T^7)-695 (T^8)+54 (T^9)-695 (T^10)+2731 (T^11)-2942 (T^12)-2988 (T^13)+10899 (T^14)-12510 (T^15)+7632 (T^16)-2520 (T^17)+360 (T^18)-12 (T^6) (Subscript[ca, 3])+24 (T^7) (Subscript[ca, 3])-12 (T^8) (Subscript[ca, 3])-12 (T^10) (Subscript[ca, 3])+24 (T^11) (Subscript[ca, 3])-12 (T^12) (Subscript[ca, 3]))

Solve[z==(T^(1/2))-(T^(-1/2)),T]

Collect[
ExpandAll[pol/.Solve[z==(T^(1/2))-(T^(-1/2)),T][[1]]],
ϵ,Together]

pol1=Series[Log[pol],{ϵ,0,Exponent[pol,ϵ]}]
Collect[
ExpandAll[pol1/.Solve[z==(T^(1/2))-(T^(-1/2)),T][[1]]],
ϵ,Together]

Collect[
Expand[pol]/.{(T^(p_.))/;p>0:>(((((2+(z^2)-z (Sqrt[4+(z^2)]))/2)))^p),(T^(p_.))/;p<0:> (((((2+(z^2)+z (Sqrt[4+(z^2)]))/2)))^(-p))},
ϵ,Expand]

T2z[p_]:=Module[{P=Expand[p],n,c},
If[P===0,0,c=Coefficient[P,T,n=Exponent[P,T]];c (z^(2n))+T2z[P-c((((T^(1/2))-(T^(-1/2))))^(2n))]]];
T2z[(1/(T^16))(((-1+T))^2) (5-18 T+33 (T^2)-32 (T^3)+2 (T^4)+42 (T^5)-62 (T^6)-8 (T^7)+166 (T^8)-242 (T^9)+108 (T^10)+132 (T^11)-226 (T^12)+148 (T^13)-11 (T^14)-36 (T^15)-11 (T^16)+148 (T^17)-226 (T^18)+132 (T^19)+108 (T^20)-242 (T^21)+166 (T^22)-8 (T^23)-62 (T^24)+42 (T^25)+2 (T^26)-32 (T^27)+33 (T^28)-18 (T^29)+5 (T^30))]